Sep 10, 2013 - the market for the E-mini Standard & Poors 500 stock index futures contract (henceforth, E-mini ). ..
Exploratory Trading ∗
Adam D. Clark-Joseph
September 10, 2013
Abstract Using comprehensive, account-labeled message records from the E-mini S&P 500 futures market, I examine the mechanisms that enable high-frequency traders (HFTs) to pro tably anticipate price movements. HFTs in my sample consistently lose money on numerous small orders, then earn high pro ts on larger orders, and I model how a trader could use her own small, exploratory orders to gather valuable, private information. Direct empirical tests of the model's predictions support the hypothesis that HFTs use this sort of exploratory trading to actively learn about likely future price movements. ∗ University of Illinois Urbana-Champaign, E-mail:
[email protected]. I thank Celso Brunetti, Richard Haynes, Andrei Kirilenko, Todd Prono, Tugkan Tuzun, Brian Weller and Steve Yang for their assistance with the empirical component of this paper, and I thank John Campbell, Brock Mendel, Andrei Shleifer, Alp Simsek and Jeremy Stein for their invaluable advice and comments. I thank seminar participants at Harvard, UBC Sauder, Indiana Kelley, Stanford GSB, Illinois, OSU Fisher, Dartmouth Tuck, Wharton, and the NYU Stern Microstructure Meeting for their useful feedback, and I thank Terrence Hendershott for his insightful discussion. I gratefully acknowledge the support of an NSF Graduate Research Fellowship. The views expressed in this paper are my own and do not constitute an o cial position of the Commodity Futures Trading Commission, its Commissioners, or sta.
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1 Introduction Automation pervades modern nancial markets, and trading responsibilities that once fell on human agents have been delegated increasingly to computer algorithms.
Much of the algorithmic activity
in major markets emanates from so-called high-frequency traders (HFTs). Although it dominates modern nancial exchanges, HFTs' activity remains largely mysterious and opaqueit is the dark matter of the trading universe. The prominence, novelty, and opacity of HFTs' activity has fueled heated public controversy about the desirability of high-frequency trading. Advocates assert that HFTs improve price discovery, liquidity, and market resilience.
Critics contend that HFTs make no unique contribution to price
discovery, provide only illusory liquidity, harm traditional market participants, and increase market fragility. This debate has sparked a variety of proposals for policies to regulate or deter high-frequency trading, and a few proposed policies have begun to be adopted. However, the impact of high-frequency trading on dierent aspects of market quality and on the welfare of non-HFT market participants is not well understood. Understanding and explaining the impacts of high-frequency trading requires knowledge of why and how HFTs' strategies work, as does the counterfactual analysis necessary for optimal policy design. To this end, the present paper analyzes economic underpinnings of HFTs' strategies in the context of the market for the E-mini Standard & Poors 500 stock index futures contract (henceforth, E-mini). I nd that HFTs in the E-mini market earn roughly
40% of their pro ts from the transactions that they
initiatethat is, from their so-called aggressive ordersand I examine the mechanism underlying HFTs' capacity to earn these pro ts.
1.1
Empirical Research on High-Frequency Training
The archetypal HFT generates a large number of trades, but this eponymous high trading frequency is only one of several traits that characterize HFTs. Two further hallmarks of HFTs are their extremely short time-frames for maintaining positions, and their propensity to end the trading day holding at most minimal net inventory positions. HFTs are also characterized by their ability to react to market events with only a tiny delay, or latency, typically on the order of milliseconds. Empirical study of high-frequency trading has proven challenging, but not impossible. ample, Brogaard
et al.
For ex-
(2012) obtain and analyze a NASDAQ dataset that ags messages from an
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aggregated group of 26 HFT rms, and Hasbrouck and Saar (2011) conduct a complementary analysis by statistically reconstructing strategic runs of linked messages in public NASDAQ data. Both of these studies suggest bene cial eects from HFTs' activity, in line with the ndings for general algorithmic trading by Hendershott
et al.
(2011) and Hendershott and Riordan (2009); however, inherent
limitations of the underlying data restrict these studies' scope to explain how and why such eects arise. Empirical results in Baron
et al.
(2012), Hagströmer and Nordén (2013), and the present paper
indicate that HFTs exhibit considerable heterogeneityeven in a market for a single asset.
As a
result, aggregate HFT activity reveals little about what individual HFTs really do. Data suitable for the study of individual HFTs' activity are di cult to obtain. Whereas publicly available 13-F forms reveal the behavior of institutional investors at a quarterly frequency, there is no comparable public data that can be used to track and analyze the behavior of individual traders at a second- or millisecond-frequency. The only fully adequate data available for academic research on high-frequency trading in U.S. markets come from regulatory records that the Chicago Mercantile Exchange provides to the U.S. Commodity Futures Trading Commission.
Kirilenko
et al.
(2010)
pioneered the use of transaction data from these records to investigate high-frequency trading in the E-mini market during the so-called Flash Crash of May 6, 2010.
This work introduced a data-
driven scheme to classify trading accounts using simple measures of overall trading activity, intraday variation in net inventory position, and inter-day changes in net inventory position. Of the accounts with su ciently small intra- and inter-day variation in net position, Kirilenko the highest levels of trading activity as HFTs
,
et al.
classify those with
and these accounts are quintessential high-frequency
traders. Kirilenko
et al.
nd that HFTs participate in over one-third of the trading volume in the E-mini
market, and subsequent research by Baron
et al.
(2012) documents the large and stable pro ts that
HFTs in the E-mini market earn. This work provides empirical con rmation of HFTs' importance, and it oers some crisp descriptions of HFTs' activity. However, it does not attempt to explain why HFTs act as they do, or how HFTs earn pro ts. Indeed, no extant empirical research attempts such explanations. In this paper, I address a central aspect of this open problem, namely the question of how HFTs in the E-mini market earn pro ts on their aggressive orders. When a trader executes an aggressive order, he pays his counterparty a fraction of the bid-ask spread, so the trader must correctly anticipate a price movement to pro t from his aggressive order. If a trader pro ts on an aggressive order, the
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standard explanation is that he was trading on some private information. However, since the sources of private information available to HFTs are unclear, this standard explanation of aggressive trading pro ts simply raises the new question of how HFTs might obtain valuable private information.
1.2
Active Learning, Exploration, and Private Information
How do HFTs in the E-mini market make money from their aggressive orders? One possibility is that HFTs merely react to public information faster than everyone else; this premise underlies the models of Jarrow and Protter (2011), Cespa and Foucault (2008), and Biais
et al.
(2010). A second possibility
is that HFTs simply front-run coming demand when they can predict future aggressive orders. HFTs
can
react to public information faster than some other market participants, and I nd that many
HFTs
do
trade ahead of predictable future aggressive orders at certain times, consistent Hirschey's
(2011) results from NASDAQ data. However, I nd evidence that the HFTs who pro t on average from their aggressive orders also do something more to obtain private information: they use a form of active learning. The concept of active learning in nancial markets dates back at least to the work of Leach and Madhavan (1992) and (1993), so these HFTs are not doing something completely novel, but rather are using a new incarnation of an old idea. I identify the HFTs who pro t from their aggressive orders, then I investigate how they manage to do this. I show that the HFTs who pro t from their aggressive trading use small aggressive orders to obtain private information about the price-elasticity of supply, which helps to forecast the price-impact 1
of predictable demand innovations.
Demand innovations in the E-mini market are easy to predict,
but the price-elasticity of supply is not, and price-impact is usually too small for indiscriminate frontrunning of predictable demand to be pro table. However, the private information about price-impact that an HFT's small aggressive orders generate enables the HFT to trade ahead of predictable demand at only those times when it is pro table to do so, i.e., when price-impact is large. To elucidate how this active-learning process works, I develop a theoretical model of what I term exploratory trading. Fundamentally, the model of exploratory trading rests on the notion that an HFT's aggressive orders generate valuable private information, speci cally, information about the price-impact of the aggressive orders that may follow. When an HFT places an exploratory order and observes a large
1 To
be more precise, the market response to these small aggressive orders provides private information about the
dynamic behavior of passive orders resting in the orderbook (which is analogous to supply elasticity). This information helps to forecast the price-impact of future aggressive order- ow (which is analogous to demand innovations). The labels supply and demand are merely heuristics in the context of the market for E-mini contracts, but they are rigorously accurate in the context of the market for liquidity/immediacy.
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price-impact, he learns that supply is temporarily inelastic. If the HFT knows that there is going to be more demand soon thereafter, he can place a larger order (even with a high price-impact) knowing that the price-impact from the coming demand will drive prices up further and ultimately enable him to sell at a premium that exceeds the price-impact of his unwinding order.
When an HFT knows
that supply is temporarily inelastic, he follows a routine demand-anticipation strategy. The purpose of exploratory trading is not to learn about future demand, but rather to identify the times at which trading in front of future demand will be pro table. In section 2, I present a model to formalize the concept of exploratory trading, and I derive the model's central testable predictions, which provide the blueprint for my empirical analysis. Using novel electronic message data at the Commodity Futures Trading Commission, I examine the pro tability of individual HFTs' aggressive orders. I nd that eight of the 30 HFTs in my sample pro t from their aggressive trading overall and signi cantly outperform non-HFTs. However, these same eight HFTs all lose money on their smallest aggressive orders.
(For brevity, I refer to these
eight HFTs as A-HFTs, and to the remaining 22 as B-HFTs.) Exploratory trading would produce just such a pattern of incurring small losses on exploratory orders then realizing large gains; these descriptive results both motivate further tests and suggest the A-HFTs' small aggressive orders as natural candidates for potential exploratory orders. To explicitly test the predictions of the exploratory trading model for the eight A-HFTs, I rst examine the extent to which information about the changes in the orderbook following small aggressive orders explains the pro ts that various traders earn on subsequent aggressive orders.
The
exploratory trading model predicts that the changes following an A-HFT's small aggressive order provide information that will explain a signi cant additional component of the A-HFT's subsequent performance, but that this information will not explain any additional component of other traders' subsequent performance. Consistent with these two predictions, I nd that the orderbook changes immediately following A-HFTs' small aggressive orders provide signi cant additional explanatory power for the respective A-HFTs' performance on their larger aggressive orders, but not for other traders' performance. I next test the prediction that the private information from the orderbook response to a given A-HFT's small aggressive orders helps to explain the
incidence
of subsequent larger aggressive
orders, in addition to explaining those orders' performance. This third prediction, too, is borne out empirically. Finally, to assess the practical importance and potential value of exploratory information, I construct a simple, purely aggressive trading strategy based strictly upon public information, then I
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construct additional trading strategies that augment the public-information strategy with crude forms of exploratory information. Using market data, I simulate the performance of these strategies and nd that incorporating exploratory information improves strategy performance dramatically, and indeed delivers simulated performance su cient to earn positive pro ts from aggressive trading. The remainder of this paper is organized as follows:
Section 2 presents a simple model of ex-
ploratory trading, along with the model's central predictions, and establishes the empirical agenda. Section 3 describes the dataset, presents some summary statistics, and precisely de nes HFTs. Section 4 addresses the overall pro tability of HFTs' aggressive orders and precisely characterizes the A-HFTs, then examines the A-HFTs' losses on small aggressive orders. Section 5 presents direct empirical tests of the exploratory trading model's key predictions, and section 6 examines the practical signi cance of exploratory information. Section 7 discusses extensions and implications of the empirical results. Section 8 concludes.
2 Exploratory Trading: Theory As noted in the introduction, demand innovations in the E-mini market are easy to predict from public market data, but the price-elasticity of supply is not. Although there are times when supply is unaccommodating and high future demand forecasts price changes that are large enough to pro t from, such times are di cult or impossible to identify using only passively gathered public market data.
In this type of setting, a trader can obtain additional information about supply conditions
by placing an exploratory aggressive order and observing how prices and supply respond.
The
additional exploratory information enables the trader to determine whether supply is accommodating (and expected price-impact small) or unaccommodating (and expected price-impact large), and this helps the trader to decide whether he can pro t by trading ahead of an imminent demand innovation. To codify the intuitions above, I use a two-period model with two possible states for supply conditions (accommodating or unaccommodating), and three possible demand innovations in the second period (positive, negative, or zero). This model should be interpreted as a single instance of the hundreds or thousands of similar scenarios that arise throughout the trading day.
To model a
market in which demand is easy to predict, but supply elasticity is not, I assume that the demand innovation is automatically revealed before it arrives in the second period, but the state of supply conditions is only revealed if a trader places an aggressive order in the rst period.
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I consider the problem facing a single trader, the HFT. In the rst period, the HFT has the opportunity to place an aggressive order and thereby learn about supply conditions. In the second period, regardless of what happened in the rst period, the HFT observes a signal about future demand, after which he again has an opportunity to place an aggressive order. The signal of future demand forecasts price innovations much more accurately when combined with information about supply conditions than it does when used on its own. If the HFT places an aggressive order in the rst period, he eectively buys supply information that he can use in the second period to better decide whether he should place another aggressive order. The rest of this section is devoted to formally developing a model of exploratory trading and extracting the model's key testable predictions. Appendix A contains full mathematical details.
2.1
Preliminaries
In an order-driven market, every regular transaction is initiated by one of the two executing transactors. The transactor who initiates is referred to as the aggressor, while the opposite transactor is referred to as the passor. The passor's order was resting in the orderbook, and the aggressor entered a new order that executed against the passor's preexisting resting order. Assuming that prices are discrete, the lowest price of any resting sell order in the orderbook (best ask) always exceeds the highest price of any resting buy order in the book (best bid) by at least one increment (the minimal price increments are called ticks).
A transaction initiated by the seller executes at the best bid,
while a transaction initiated by the buyer executes at the best ask; the resulting variation in transaction prices between aggressive buys and aggressive sells is known as bid-ask bounce.
Hereafter,
except where otherwise noted, I will restrict attention to price changes distinct from bid-ask bounce. Empirically, the best ask for the most actively traded E-mini contract almost always exceeds the best bid by exactly one tick during regular trading hours, so movements of the best bid, best ask, and mid-point prices are essentially interchangeable. If the best bid and best ask were held xed, a trader who aggressively entered then aggressively exited a position would lose the bid-ask spread on each contract, whereas a trader who passively entered then passively exited a position would earn the bid-ask spread on each contract. Intuitively, aggressors pay for the privilege of trading precisely when they wish to do so, and passors are compensated for the costs of supplying this immediacy, cf. Grossman and Miller (1988). These costs include xed operational costs and costs arising from adverse selection.
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Cf.
Glosten and Milgrom (1985), Stoll
(1989). An aggressive order will execute against all passive orders at the best available price level before executing against any passive orders at the next price, so an aggressive order will only have a literal price-impact if it eats through all of the resting orders at the best price. In the E-mini market, it is rare for an aggressive order to have a literal price-impact, not only because there are typically enormous numbers of contracts at the best bid and best ask, but also because aggressive orders overwhelmingly take the form of limit orders priced at the opposite best (which cannot execute at the next price level).
2.2
Baseline Model
Let time be discrete, consisting of two periods, discrete prices.
t = 1, 2,
and consider an order-driven market with
Assume that both the orderbook and order- ow are observable.
Conceptually, the
ow of aggressive orders is analogous to demand, while the set of passive orders in the orderbook (resting depth) is analogous to supply. Empirically, in the E-mini market, the sign (buy vs. sell) of aggressive orders is highly persistent, so the signs of future aggressive orders are very easy to forecast.
2.2.1
The HFT
Consider a single trader, the HFT, who has the opportunity to submit an aggressive order at the start of each time-period. The HFT submits only aggressive orders, and these orders are limited in size to
N
contracts or fewer. Let
qt ∈ {−N, . . . , −1, 0, 1, . . . , N }
aggressive order that the HFT places in period
t,
denote the signed quantity of the
where a negative quantity represents a sale, and
a positive quantity represents a purchase. Assume that the HFT only trades contracts at the initial best bid/ask, so his orders aect resting depth in the orderbook but have no literal price-impact. Suppose that prices remain constant between periods 1 and 2, and denote the price-change at the end of period
2
by
y ∈ {−1, 0, +1}.
Assume that the HFT pays constant trading costs of
per contract, and that the HFT's pro t from the aggressive order he places in period
πt = yqt − α |qt |
The lower bound of upper bound of
1
0.5
on
α
t
α ∈ (0.5, 1)
is given by
(1)
corresponds to half of the minimum possible bid-ask spread, while the
merely excludes trivial cases of the model in which aggressive orders are always
unpro table for the HFT.
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Denote the HFT's total combined pro ts from periods 1 and 2 by
πtotal := π1 + π2
(2)
Assume that the HFT is risk-neutral and seeks to maximize the expectation of
2.2.2
πtotal .
Passive Orders
There are two possible states for the behavior of passive orders: accommodating and unaccommodating. Let the variable
Λ
dating liquidity state by
represent this state, which I call the liquidity state. Denote the accommo-
Λ = A,
same in both time-periods, probability
and the unaccommodating state by
t = 1, 2.
Assume that
Λ=U
with
Λ = U.
ex-ante
The liquidity state is the
probability
u,
and
Λ=A
with
1 − u.
Intuitively, aggressive orders have a small price-impact in the accommodating liquidity state, and a large price-impact in the unaccommodating liquidity state.
The liquidity state characterizes the
behavior of resting depth in the orderbook after an aggressive order executesa generalization of price-impact appropriate for an order-driven market. When an aggressive buy (sell) order executes, it mechanically depletes resting depth on the sell (buy) side of the orderbook. Following this mechanical depletion, traders may enter, modify, and/or cancel passive orders, so resting depth at the best ask (bid) can either replenish, stay the same, or deplete further. The aggressive order's impact is oset to some extentor even reversedif resting depth replenishes, whereas the aggressive order's impact is ampli ed if resting depth depletes further. In the accommodating state replenishes, while in the unaccommodating state
(Λ = U )
(Λ = A) resting depth weakly
resting depth further depletes.
Although the orderbook is assumed to be observable, the static features of passive orders in the orderbook do not directly reveal the liquidity state
Λ.
Because the liquidity state relates to the
dynamic behavior of resting depth after an aggressive order executes,
Λ can only be deduced from the
changes in the orderbook that follow the execution of an aggressive order. For the baseline model, I will assume that the HFT learns
Λ
prior to period 2 if and only if he places an aggressive order in
period 1; I relax this assumption in section 2.4.
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2.2.3
Aggressive Order-Flow
At the end of period 2, traders other than the HFT exogenously place aggressive orders.
ϕ ∈ {−1, 0, +1}
variable
characterize this exogenous aggressive order- ow.
does not depend on the liquidity state,
ϕ = +1
and
probability
ϕ = −1
1 − v.
Λ,
The realization of
ϕ
nor does it depend on the HFT's actions; assume that
P {ϕ = +1} = P {ϕ = −1} = v/2,
with equal probabilities
The variable
Let the
and
ϕ = 0
with
ϕ is just a coarse summary of the order- owit does not represent the
actual number of contracts. Intuitively,
ϕ = −1
represents predictable selling pressure and
ϕ=0
represents predictable buying pressure, while
ϕ = +1
represents an absence of predictable pressure in
either direction.
2.2.4
Price Change
Together, the exogenous aggressive order- ow
ϕ and the liquidity state Λ determine y , the price-change
at the end of period 2, as follows:
y=
When the liquidity state is the price, and
y = ϕ.
0
if Λ = A
(3)
However, if the liquidity state is accommodating
Note that because
y=0
even when
E [y|Λ = U ] = E [ϕ] = 0
is zero, as is the period-1 expectation of
2.2.5
if Λ = U
un accommodating (Λ = U ), the exogenous aggressive order- ow can aect
does not aect the price, and
y
ϕ
(Λ = A),
aggressive order- ow
ϕ 6= 0. and
E [y|Λ = A] = 0,
the unconditional expectation of
y.
Model Timeline
Period 1
In period 1, the HFT has the opportunity to submit an aggressive order and then observe
any subsequent change in resting depth. The HFT cannot observe the liquidity state directly, but he can infer the value of conclude that
Λ=U
Λ
from changes in resting depth if he places an aggressive order; the HFT can
if resting depth further depletes following his order, and that
If the HFT does not place an aggressive order in period 1, he does not learn
Period 2
Λ=A
otherwise.
Λ.
At the start of period 2, the HFT observes the signal of future aggressive order- ow,
The HFT observes
ϕ
ϕ.
regardless of whether he placed an aggressive order in period 1 (this re ects the
idea that aggressive order- ow is easy to predict on the basis of public market data). After the HFT
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observes
ϕ,
he once again has an opportunity to place an aggressive order. Finally, after the HFT has
the chance to trade, aggressive order- ow characterized by by
ϕ
Λ
and
2.3
ϕ arrives,
and prices change as determined
in equation (3).
Analysis of the Baseline Model
The baseline model of exploratory trading makes the concept of exploratory trading more precise and transparent, and more importantly, it delivers testable implications of the hypothesis that a given trader engages in exploration.
2.3.1
When
Solving for the HFT's Optimal Trading Strategy
α > u,
the HFT will never place an order in period 2 if he doesn't know the liquidity state,
and I focus on this case to simplify the exposition; results are qualitatively unchanged for
u≥α
(see
Appendix A). I solve for the HFT's optimal trading strategy via backward induction.
Period 2
If the HFT learned the liquidity state during period 1, his optimal aggressive order in
period 2 will depend on the values of both
q2 = ϕN
is to set then
Λ,
if
Λ = U,
and to set
ϕ
q2 = 0
and if
Λ.
The HFT's optimal strategy when he knows
Λ=A
. Taking expectations with respect to
ϕ
Λ
and
we nd
E [π2 |Λ known]
∗
= N v (1 − α) u + 0∗ (1 − u)
(4)
= N vu (1 − α)
If the HFT did not learn the liquidity state during period 1, his (constrained) optimal aggressive order in period 2 will still depend on the value of rather that the actual value of
q2 = ϕN
when
u ≥ α,
Λ.
and to set
ϕ,
but it will only depend on the
The HFT's optimal strategy when he does not know
q2 = 0
when
α > u.
I assumed for simplicity that
E [π2 |Λ unknown] = 0
Period 1
distribution
At the start of period 1, the HFT knows neither
α > u,
Λ
of
Λ,
is to set
so
(5)
ϕ
nor
Λ,
but he faces the same trading
costs (α per contract) as in period 2. Consequently, the HFT's expected direct trading pro ts from a
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period-1 aggressive order are negative, given by
E [π1 ]
= −α |q1 |
(6)
Since there is no noise in this baseline model, and the HFT learns
Λ
perfectly from any aggressive
order that he places in the rst period, we can restrict attention to the cases of
q1 = 0
and
|q1 | = 1.
We obtain the following expression for the dierence in the HFT's total expected pro ts if he sets
|q1 | = 1
instead of
q1 = 0:
E [πtotal | |q1 | = 1] − E [πtotal |q1 = 0] = N vu (1 − α) − α
The HFT engages in exploratory trading if he sets ing if he sets
q1 = 0,
(7)
|q1 | = 1, and he does not engage in exploratory trad-
so equation (7) represents the expected net gain from exploration. Exploratory
trading is optimal for the HFT when this expected net gain is positive.
2.3.2
Order-Sizes and Conditions for Exploratory Trading
The results in section 2.3.1 demonstrate the trade-o between direct trading costs and informational gains at the heart of exploratory trading. By placing a (costly) aggressive order in period 1, the HFT buys the perturbation needed to elicit a response in resting depth that reveals the liquidity state. Knowing the liquidity state enables the HFT, in period 2, to better determine whether placing an aggressive order will be pro table. Parameters of the model determine the relative costs and payos of exploration. I derive routine comparative statics in Appendix A, but because exogenous variation in
N, v , α,
or
u
is scarce, these
comparative statics provide little in the way of testable implications. However, for the HFT to weakly prefer to engage in period-1 exploratory trading with order-size
q1 ≥ 1,
it follows almost immediately
from (7) that we must have
N vu (1 − α) − q1 α ⇐⇒ N
≥ 0 ≥
α 1−α
(8)
1 vu
q1
(9)
The small exploratory trade/large follow-up trade pattern in the baseline model is more than just an artifact of the simplifying assumption that the size of the exploratory order didn't aect the infor-
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mativeness of the subsequent orderbook response. Rather, the pattern arises because the exploratory orders are always costly in expectation, while the resulting exploratory information is only valuable when there is predictable aggressive order- ow in the next period (i.e., when
ϕ 6= 0).
The per-contract
losses on exploratory orders will therefore be greater in magnitude than the per-contract pro ts on follow-up orders, so the total pro ts on follow-up orders will only exceed the total losses on exploratory orders if the follow-up orders are larger. The central goal of this paper is not to explain why pro table aggressive orders might tend to be large and unpro table ones small, but rather to understand how an HFT could consistently pro t from his aggressive trading at all. Although a pattern of losses on small orders and more-than-osetting gains on larger orders falls naturally out of the baseline exploratory trading model, this is not at all the main substance or purpose of the model. The distinctive predictions of the exploratory trading model relate to the information revealed by the market response to exploratory orders.
2.3.3
Testable Predictions
The baseline exploratory trading model generates two key testable implications of the hypothesis that a given trader engages in exploration. Assume for the moment that candidate exploratory orders can be distinguished from the trader's other orders (I discuss this assumption in section 2.5). First, the model predicts that the market response following an exploratory order helps to forecast whether or not prices will change soon thereafter, and also whether or not the exploring trader will place a follow-up order in the direction of an imminent price-change. Hence the model implies that
if
a given trader engages in exploration, then the market response to his exploratory orders should help to explain his earnings on subsequent aggressive orders. Next, the baseline exploratory trading model predicts that the market response following an exploratory order helps to forecast whether or not the explorer will place a follow-up order. From the results in section 2.3.2, we know that the follow-up order must tend to be larger than the exploratory order, so the model implies that
if a given trader engages in exploration, then the market response to
his exploratory orders should help to explain the incidence of his larger aggressive orders. 2.4
Private Gains from Exploratory Trading
The baseline model developed in the previous section abstracted away from the details of the HFT's inference about
Λ, and it made the simplifying assumption that placing an aggressive order in the rst
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period was both necessary and su cient for the HFT to learn the liquidity state. This simpli cation does not qualitatively aect the two testable implications highlighted in section 2.3, but the necessity assumption obscures why the HFT learns more from placing an aggressive order himself than he does from merely observing an aggressive order placed by someone else.
Relaxing this necessity
assumption delivers a third, important, testable implication. Sometimes, the changes in the orderbook following the arrival of an aggressive order are truly a response caused by the aggressive order. Often, though, both the aggressive order and the subsequent orderbook activity are really just common responses to some third event, and there is no causal link between the aggressive order and the orderbook changes that occur after its arrival. Factors other than aggressive order arrivals can aect the behavior of resting depth. In particular, a trader may adjust her passive orders in response to new information. Just as a trader might place an aggressive buy order if he believes that prices are too low, so might another trader who shared this belief cancel some of her passive sell orders. As a result, changes in resting depth are typically correlated with aggressive order- ow, even when the aggressive orders do not actually cause those changes. Changes in resting depth
caused
by aggressive orders help to forecast the price impact of future aggressive order- ow,
while changes in the orderbook correlated with an aggressive order's arrivalbut not caused by itdo not help. If someone else placed the aggressive order, these two scenarios are indistinguishable to the HFT, so the possibility that he is observing the uninformative non-causal scenario attenuates the amount that he can learn from the market response to someone else's aggressive order. By contrast, if the HFT places an exploratory order himself, he can be entirely sure that he did so for exogenous reasons, so the uninformative scenario is not a concern. The HFT learns more from his own aggressive orders than he does from those of traders because he can better infer causal eects from aggressive orders that he himself placed.
2.4.1
Formalizing the Intuition
To make the preceding intuition more rigorous, consider a variant of the baseline model from section 2.2 in which someone other than the HFT places an aggressive order at the beginning of period 1. With probability
ρ,
this aggressive order is the result of an unobservable informational shock, and
resting depth further depletes following the order, regardless of the liquidity state probability
1 − ρ)
unaccommodating.
Λ.
Otherwise (with
resting depth further depletes after the order if and only if the liquidity state is Aside from this new aggressive order, all other aspects of the baseline model
14
remain unchanged. If the HFT places an aggressive order in period 1, his expected total pro ts are the same as they were in the baseline model, i.e.,
E [πtotal | |q1 | = 1]
= N vu (1 − α) − α
(10)
However, the HFT's expected pro ts if he does not place an order in period 1 are now higher than they were in the baseline model, because the HFT learns something from the depth changes following the other trader's aggressive order. If resting depth weakly replenishes after that order, the HFT learns with certainty that the liquidity state is accommodating (i.e.,
Λ = A),
so the HFT will not submit an
aggressive order in period 2, and his total pro ts will be zero. Alternatively, if resting depth further depletes following the other trader's aggressive order, we have
P {Λ = U |resting depth f urther depletes}
The HFT's optimal strategy when he does not know set
q2 = 0
Λ
is to set
otherwise. Taking expectations with respect to
Λ
u u + ρ (1 − u)
=
q2 = ϕN
and
ϕ,
when
u u+ρ(1−u)
(11)
≥ α,
we nd that the HFT's
and to
ex-ante
expected total pro ts in this case are given by
E [πtotal |AO by someone else] = max N v
2.4.2
u − α ,0 u + ρ (1 − u)
(12)
Analysis and Additional Testable Implication
The features of the baseline model discussed in section 2.3 are qualitatively unchanged in the modi ed version, but now the privacy parameter
ρ
also exerts an in uence. In the limiting case where the
depth change following an aggressive order placed by someone else is completely uninformative to the HFT (i.e.,
ρ = 1),
equation (12) collapses down to equation (5) from the baseline model.
At the
opposite extreme, when the HFT learns the liquidity state perfectly from observing another trader's aggressive order (i.e.,
ρ = 0),
the HFT's expected total pro ts are unambiguously lower if he places
an aggressive order in period 1 himself. through mere observation, as he can when
When the HFT can learn more about the liquidity state
ρ
is smaller, he has less incentive to incur the direct costs
of exploratory trading. Viewed dierently, if the HFT
does
nd it optimal to engage in exploratory trading, it must be
15
the case that he obtains more useful information from the market response to his aggressive orders than he does from the market response to other traders' aggressive orders. In an anonymous market, it also follows from symmetry that each other trader obtains no more useful information from the market response to the HFT's aggressive orders than they do from the market response to another arbitrary trader's aggressive orders. This provides a third testable implication of the hypothesis that a given trader engages in exploration.
2.5
Empirical Agenda
Before attempting any empirical evaluation of the hypothesis that the HFTs who earn positive pro ts on average from their aggressive orders engage in exploratory trading, two basic issues must be addressed. First, it must be determined which HFTs, if any, actually earn positive and abnormal pro ts from their aggressive trading. I address this matter in section 4.2, and I identify eight such HFTs, to whom I refer as A-HFTs.
Next, among the A-HFTs' aggressive orders, suitable candidates for
putative exploratory orders must be identi ed in some manner. The results from section 2.3.2 suggest that small, unpro table aggressive orders are prime candidates. In section 4.4, I nd that all of the A-HFTs, indeed, tend to lose money on their smallest aggressive orders, consistent with the theory that these orders are placed for exploratory ends. With these two preliminary matters resolved, I turn to direct empirical tests of the model's key predictions. As a benchmark, I consider the market response following the last small aggressive order placed by anyone, which is public information. The empirical implications discussed earlier in this section can then be condensed into three central predictions, namely that relative to the publicinformation benchmark, information from the market response following an A-HFT's small aggressive orders:
Predict.1. Explains a significant additional component of that A-HFT's earnings on subsequent aggressive orders, but
Predict.2. Does not explain any additional component of other traders' earnings on subsequent aggressive orders, and
Predict.3. Further explains by a significant margin the incidence of that A-HFT's subsequent large aggressive orders
In section 4.3, I make rigorous the notion of explaining earnings on subsequent aggressive orders,
16
then in section 5, I introduce an explicit numeric measure of market response and formally test the predictions above.
3 High-Frequency Trading in the E-mini Market The E-mini S&P 500 futures contract is a cash-settled instrument with a notional value equal to $50.00 times the S&P 500 index. Prices are quoted in terms of the S&P 500 index, at minimum increments, ticks, of
0.25
index points, equivalent to
$12.50
per contract. E-mini contracts are created directly
by buyers and sellers, so the quantity of outstanding contracts is potentially unlimited. All E-mini contracts trade exclusively on the CME Globex electronic trading platform, in an orderdriven market. Transaction prices/quantities and changes in aggregate depth at individual price levels in the orderbook are observable through a public market-data feed, but the E-mini market provides full anonymity, so the identities of the traders responsible for these events are not released.
Limit
orders in the E-mini market are matched according to strict price and time priority; a buy (sell) limit order at a given price executes ahead of all buy (sell) limit orders at lower (higher) prices, and buy (sell) limit orders at the same price execute in the sequence that they arrived. Certain modi cations to a limit order, most notably size increases, reset the time-stamp by which time-priority is determined. E-mini contracts with expiration dates in the ve nearest months of the March quarterly cycle (March, June, September, December) are listed for trading, but activity typically concentrates in the contract with the nearest expiration.
Aside from brief maintenance periods, the E-mini market is
open 24 hours a day, though most activity occurs during regular trading hours, namely, weekdays between 8:30 a.m. and 3:15 p.m. CT.
3.1
Description of the Data
I examine account-labeled, millisecond-timestamped records at the Commodity Futures Trading Commission of the so-called business messages entered into the Globex system between September 17, 2010 and November 1, 2010 for all E-mini S&P 500 futures contracts. This message data captures not only transactions, but also events that do not directly result in a trade, such as the entry, cancellation, or modi cation of a resting limit order. Essentially, business messages include any action by a market participant that could potentially result in or aect a transaction immediately, or at any point
17
in the future.
2
I restrict attention to the December-expiring E-mini contract (ticker ESZ0). During
my sample period, ESZ0 activity accounted for roughly contracts, and more than
99.9%
98%
of the message volume across all E-mini
of the trading volume.
The price of an ESZ0 contract during this period was around trading volume averaged
1, 991, 252
averaged approximately
5
contracts or approximately
$55, 000
$115
to
$60, 000,
and (one-sided)
billion per day. Message volume
million business messages per day, and the number of aggressive orders
executed per day day averaged
132, 127.
The intensity of trading varies considerably throughout the
day (aggressive orders typically arrive in tight clusters), so the median time interval between aggressive orders during regular trading hours is closer to
200
20 milliseconds than it is to the mean interval of roughly
milliseconds.
3.2
De ning High-Frequency Traders
Kirilenko
et al.
identify as HFTs those traders who exhibit minimal accumulation of directional posi-
tions, high inventory turnover, and high levels of trading activity. I, too, use these three characteristics to de ne and identify HFTs. To quantify an account's accumulation of directional positions, I consider the magnitude of changes in end-of-day net position as a percentage of the account's daily trading volume. Similarly, I use an account's maximal intraday change in net position, relative to daily volume, to measure inventory turnover. Finally, I use an account's total trading volume as a measure of trading activity. I select each account whose end-of-day net position changes by less than 6% of its daily volume, and whose maximal intraday net position changes are less than 20% of its daily volume. I rank the selected accounts by total trading volume, and classify the top 30 accounts as HFTs. The original classi cations of Kirilenko
et al.
and Baron
et al.
guided the rough threshold choices for inter-day
and intraday variation. Thereafter, since con dentiality protocols prohibit disclosing results for groups smaller than eight trading accounts, the precise cuto values of
6%, 20%, and 30 accounts were chosen
to ensure that all groups of interest would have at least eight members. My central results are not sensitive to values of these parameters. The set of HFTs corresponds closely to the set of accounts with the greatest trading volume in my sample, so the set of HFTs is largely invariant both to the exact characterizations of inter-day
2 Excluded
from these data are purely administrative messages, such as log-on and log-out messages.
The good-
'til-cancel orders in the orderbook at the start of September 2, and a small number of modi cation messages (around
2 − 4%)
are also missing from these records. Because I restrict attention to aggressive orders, and I only look at
in resting depth (rather than its actual level), my results are not sensitive to these omitted messages.
18
changes
and intraday variation in net position relative to volume, and to the exact cuto values for these quantities. Similarly, changing the 30-account cuto to (e.g.) 15 accounts or 60 accounts does not substantially alter my results, because activity heavily concentrates among the largest HFTs.
For
example, the combined total trading volume of the 8 largest HFTs exceeds that of HFTs 9-30 by roughly three-quarters, and the combined aggressive volume of the 8 largest HFTs exceeds that of HFTs 9-30 by a factor of almost
3.3
2.5.
HFTs' Prominence and Pro tability
Although HFTs constitute less than
0.1%
of the
41, 778
accounts that traded the ESZ0 contract
between September 17, 2010 and November 1, 2010, they participate in
46.7%
of the total trading
volume during this period. In addition to trading volume, HFTs are responsible for a large fraction of message volume.
During the sample period, HFTs account for
31.9%
of all order entry, order
modi cation and order cancellation messages. The HFTs also appear to earn large and stable pro ts. Gross of trading fees, the 30 HFTs earned a combined average of
$1.51
million per trading day during
the sample period. Individual HFTs' annualized Sharpe ratios are in the neighborhood of 10 to 11. The Chicago Mercantile Exchange reduces E-mini trading fees on a tiered basis for traders whose average monthly volume exceeds various thresholds. Trading and clearing fees were either contract or
$0.12
per contract for the 20 largest HFTs, and were at most
remaining HFTs. Initial and maintenance margins were both
$4, 500
$0.16
$0.095
per
per contract for the
for all of the HFTs.
Hereafter, unless otherwise noted, I restrict attention to activity that occurred during regular trading hours. HFTs' aggressive trading occurs almost exclusively during regular trading hours (approximately
95.6%,
by volume), and market conditions during these times dier substantially from
those during the complementary o-hours.
4 HFTs' Pro ts from Aggressive Orders Aggressive trading is a tremendously important component of HFTs' activity. In aggregate, approximately
48.5% of HFTs' volume is aggressive, and this gure rises to 54.2% among the 12 largest HFTs.
Furthermore, many HFTs consistently pro t from their aggressive trading. Since the bid-ask spread in the E-mini market rarely exceeds the minimum imposed upon it by the granularity of prices, there
19
3
is little mystery about how a trader's passive trades could consistently earn money.
By contrast,
explaining how a trader who uses only market data could consistently pro t on aggressive trades is somewhat di cult.
4.1
Measuring Aggressive Order Pro tability
Because all E-mini contracts of a given expiration date are identical, it is neither meaningful nor possible to distinguish among the individual contracts in a trader's inventory, so there is generally no way to determine the exact prices at which a trader bought and sold a particular contract. As a result, it is typically impossible to measure directly the pro ts that a trader earns on an individual aggressive order. However, the cumulative price change following an aggressive order, normalized by the order's direction (+1 for a buy, or
−1
for a sell), can be used to construct a meaningful proxy
for the order's pro tability. Intuitively, the average expected pro t from an aggressive order equals the expected favorable price movement, minus trading/clearing fees and half the bid-ask spread. See Appendix B for rigorous justi cation. Estimating the cumulative favorable price movement after an aggressive order is straight-forward. Consider a trader who can forecast price movements up to
j
time periods in the future, but no further.
If the trader places an aggressive order in period t, any price changes that she could have anticipated at the time she placed the order will have occurred by period
t+j +1.
Provided that price is a martingale
with respect to its natural ltration, the expected change in price from period
t+j+1
onward is
zero, both from the period−t perspective of the trader and from an unconditional perspective. Thus the change in price between period trader's order (+1 for a buy, or
−1
t
and any period after
t + j,
normalized by the direction of the
for a sell), will provide an unbiased estimate of the favorable price
movement following the trader's order. The remarks above imply that we can derive a proxy for the pro tability of an HFT's aggressive order using the (direction-normalized) accumulated price-changes following that aggressive order out to some time past the HFT's maximum forecasting horizon. If we choose too short an accumulation window, the resulting estimates of the long-run direction-normalized average price changes following the HFT's aggressive orders will be biased downward. As a result, we can empirically determine an adequate accumulation period by calculating cumulative direction-normalized price changes over longer
3 Explaining
the pro tability of individual passive trades does not resolve the question of how various HFTs manage
to participate in so many passive trades. In equilibrium, we would expect new entrants to reduce the average passive volume of an individual trader until her total pro ts from passive trades equaled her xed costs.
20
and longer windows until their mean ceases to signi cantly increase. Using too long an accumulation period introduces extra noise, but it will not bias the estimates. I nd that an accumulation period, measured in event-time, of 30 aggressive order arrivals is su cient to obtain unbiased estimates; for all of the empirical work in this paper, I use an accumulation period of 50 aggressive order arrivals to allow a wide margin for error. See Appendix B for further details. As noted earlier, the bid-ask spread for the E-mini is almost constantly
$12.50
regular trading hours, and the HFTs in my sample face trading/clearing fees of
(one tick) during
$0.095
to
$0.16
per
contract, so the average favorable price movement necessary for an HFT's aggressive order to be pro table is between
$6.345
and
$6.41
per contract. Since trading/clearing fees vary across traders, I
report aggressive order performance in terms of favorable price movement, that is, earnings gross of fees and the half-spread.
4.2
HFTs' Overall Pro ts from Aggressive Orders
To measure the overall mean pro tability of a given account's aggressive trading, I compute the average cumulative price change following each aggressive order placed by that account, weighted by executed quantity and normalized by the direction of the aggressive order. As a group, the 30 HFTs in my sample achieve average aggressive order performance of basis, nine HFT accounts exceed the relevant
$6.25 + f ees
$7.01
per contract. On an individual
pro tability hurdle, and each of these nine
accounts exceeds this hurdle by a margin that is statistically signi cant at the
0.05
level.
One of
these nine accounts is linked with another HFT account, and their combined average performance also signi cantly exceeds the pro tability hurdle. Overall, the HFTs vastly outperform non-HFTs, who earn a gross average of
$3.19 per aggressively-
traded contract. However, these overall averages potentially confound eects of very coarse dierences in the times at which traders place aggressive orders with eects of the ner dierences more directly related to strategic choices. For example, if all aggressive orders were more pro table between 1 p.m. and 2 p.m.
than at other times, and HFTs only placed aggressive orders during this window, the
HFTs' outperformance would not depend on anything characteristically high-frequency. To control for potential low-frequency confounds, I divide each trading day in my sample into 90-second segments and regress the pro tability of non-HFTs' aggressive orders during each segment on both a constant and the executed quantities of the aggressive orders. Using these local coe cients, I compute the pro tability of each aggressive order by an HFT in excess of the expected pro tability
21
of a non-HFT aggressive order of the same size during the relevant 90-second segment. With these additional controls, only 27 HFT accounts continue to exhibit signi cant outperformance of nonHFTs, and only eight of the 27 accounts are among those whose absolute performance exceeded the pro tability hurdle.
4.2.1
A-HFTs and B-HFTs
For expositional ease, I will refer to the eight HFT accounts that make money on their aggressive trades
and
outperform the time-varying non-HFT benchmark as A-HFTs, and to the complementary set
of HFTs as B-HFTs. The eight A-HFTs have a combined average daily trading volume of
59.2%
contracts, and on average,
average daily trading volume of A-HFTs place an average of a median size of
10
of this volume is aggressive.
828, 924
8, 994
contracts, of which
earn a combined average of
35.9%
is aggressive. Together, the eight
aggressive orders per day, with a mean size of
8.3
contracts and a median size of
$793, 342
B-HFTs earn a combined average of
1
$715, 167
31, 113
$99, 168
per day, or an individual average of
$6.37
60.3
contracts and
aggressive orders
contract. Gross of fees, the A-HFTs
per day, or an individual average of
The highest pro tability hurdle among the A-HFTs is
4.3
The 22 B-HFTs have a combined
contracts. The 22 B-HFTs together place an average of
per day, with a mean size of
982, 988
per day, while the
$32, 508
per day.
4
per aggressively traded contract.
Relative Aggressive Order Pro tability: HFT vs. Econometrician
To gain some insight into the factors that aect aggressive trading pro ts, I examine the extent to which econometric price forecasts explain the realized performance of aggressive orders placed by A-HFTs, B-HFTs, and non-HFTs. The methodology that I develop in this subsection provides the starting point for my direct tests of the exploratory trading model's predictions in section 5.
4.3.1
Variables that Forecast Price Movements
Bid-ask bounce notwithstanding, the price at which aggressive orders execute changes rather infrequently in the E-mini market. On average, only about
1 − 3%
of aggressive buy (sell) orders execute
at a nal price dierent from the last price at which the previous aggressive buy (sell) order executed, and the price changes that do occur are almost completely unpredictable on the basis of past price changes. However, several other variables forecast price innovations surprisingly well.
4 All
of the preceding descriptive statistics include the small amount of trading activity that occurred outside regular
trading hours.
22
In contrast to price innovations, the direction of aggressive order- ow in the E-mini market is extremely persistent and predictable. On average, the probability that the next aggressive order will be a buy (sell) given that the previous aggressive order was a buy (sell) is around
75%.
This persistence
in the direction of aggressive orders can be interpreted as an artifact of many dierent traders trying to do the same thing at the same time, but with some distribution of reaction speeds. In addition to forecasting the direction of future aggressive order- ow, the direction of past aggressive order- ow also forecasts future price innovations to statistically and economically signi cant extent, and forecasts based on past aggressive order signs alone are moderately improved by information about the (signed) quantities of past aggressive orders. Simple measures of recent changes in the orderbook oer further, yet modest, improvement in price forecasts. The levels of resting depth in the orderbook, in addition to the changes in resting depth, also improve price forecasts slightly, but these stock variables cannot be reliably recovered in much of my dataset because a small number of modi cation messages (around
2 − 4%)
are missing. These occa-
sional missing modi cations introduce only transient noise into ow variables such as changes in resting depth, but they have permanent eects on the corresponding stock variables. Fortunately, omitting resting-depth stock variables from the direct tests of the exploratory trading model's predictions in section 5 is harmless. These tests use the explanatory variables in the benchmark regression (13) only as an empirical analogue of
ϕ,
the signal of future aggressive order- ow in the exploratory trading
model. Thus the tests require only that the benchmark explanatory variables oer some predictive power, not that those variables control for all public information (I control for public information by other means, discussed in section 5).
4.3.2
Econometric Benchmark
For each trading day in my sample, I regress the cumulative price-change (in dollars) between the aggressive orders
k
and
k + 50,
denoted
section 4.3.1. Speci cally, I regress and
k
orders
yk
yk ,
on lagged market variables suggested by the remarks in
on the changes in resting depth between aggressive orders
k−1
at each of the six price levels within two ticks of the best bid or best ask, the signs of aggressive
k − 1 through k − 4, and the signed executed quantities of aggressive orders k − 1 through k − 4.
For symmetry, I adopt the convention that sell depth is negative and buy depth is positive, so that an increase in buy depth has the same sign as a decrease in sell depth. Denoting the row vector of
23
the 14 regressors by
zk−1 ,
and a column vector of 14 coe cients by
yk
Γ,
I estimate the equation
= zk−1 Γ + k
(13)
The estimated coe cients have the expected signs, and their joint signi cance is extremely high. I discuss the regression results directly and report coe cient estimates in Appendix C. To compute the excess performance of aggressive order
signk ,
residual by
the sign of the
k th
k , denoted ξk , I normalize the k th regression
aggressive order:
ˆ ξk = signk yk − zk−1 Γ
(14)
As discussed in section 4.1, normalizing the cumulative price-change order yields a measure of the measure of
k th
yk by the sign of the k th aggressive
aggressive order's pro tability. Likewise, the quantity
ξk
provides a
k th aggressive order's pro tability in excess of that expected on the basis of the benchmark
econometric speci cation. I compute the vectors of direction-normalized residuals separately for each of the 32 trading days in my sample, then combine all of them into a single vector for the entire sample period.
4.3.3
Explained Performance
The price movements predicted by (13) explain a substantial component of the performance of ag5
gressive orders placed by A-HFTs, B-HFTs, and non-HFTs alike.
Looking ahead, this explanatory
power validates the use of speci cation (13) as a basis for the more sophisticated analyses in section 5. Figure 1 and Table 1 summarize the overall size-weighted average performance of aggressive orders placed by various trader groups, both in absolute terms, and in excess of the econometric benchmark. Con dence intervals are computed via bootstrap.
5 Although
the variables in
not literally a forecast of
yk
zk−1
ˆ is kth aggressive order arrives, the tted value zk−1 Γ ˆ is estimated from data for the entire day. However, the coe cient Γ ˆ as a forecast of yk is innocuous in sample period, so thinking of zk−1 Γ
are all observable before the
in the strictest sense, as
estimates are extremely stable throughout the the present setting. See section 6.1.
24
Gross Earnings per Contract (Dollars)
Figure 1. Aggressive Order Performance Relative to Econometric Benchmark Explained Performance
Excess Performance
9 8 7 6 5 4 3 2 1 0
A-HFTs
B-HFTs
Non-HFTs
Figure 1: Aggressive Order Performance Relative to Econometric Benchmark
Table 1: Aggressive Order Performance vs. Econometric Benchmark Mean Absolute
Mean Absolute
99%
CI
Mean Excess
Mean Excess
99%
CI
Explained Performance
A-HFTs
7.65
(7.54, 7.73)
3.22
(3.09, 3.36)
57.8%
B-HFTs
5.67
(5.57, 5.77)
2.04
(1.90, 2.17)
64.0%
Non-HFTs
3.19
(3.12, 3.26)
0.22
(0.14, 0.29)
93.06%
Although the unexplained component of the A-HFTs' performance is large in absolute terms, and massive relative that of the B-HFTs and the non-HFTs, the econometric controls in (13) nevertheless explain more than half of the A-HFTs' performance on their aggressive orders.
The exploratory
trading model developed earlier assumed that A-HFTs ultimately traded ahead of easily predictable demand innovations (when liquidity conditions were suitably unaccommodating), and the explanatory power of the econometric benchmark for the A-HFTs' performance substantiates this assumption.
25
4.4
A-HFTs' Losses on Small Aggressive Orders 6
The A-HFTs' aggressive orders tend to become more pro table as order-size increases. earning money from their aggressive orders on average, the A-HFTs all tend to
lose
In fact, despite money on the
smallest aggressive orders that they place. Note also that I refer here to the size of the aggressive orders A-HFTs submit, not the quantity that executes, so the small orders were intentionally chosen to be small, and the large orders intentionally chosen to be large. To make precise both the meaning of small aggressive orders, and A-HFTs' losses on them, I specify cutos for order-size and compute the average performance of A-HFTs' aggressive orders below and above those size cutos. Figure 2 and Table 2 display bootstrap con dence intervals for the executed-quantity-weighted average performance of A-HFTs' aggressive orders weakly below and strictly above various order-size cutos. Figure 2. A-HFT Performance on Small
Gross Earnings per Contract (Dollars)
AOs Below Cutoff Size
and Larger Aggressive Orders (95% Conf. Intervals) AOs Above Cutoff Size
Profitability Hurdle
9 8 7 6 5 4 3 2 1 0 0
5
10
15
20
Aggressive Order Cutoff Size
Figure 2: A-HFT Performance on Small and Larger Aggressive Orders (95% Conf. Intervals)
6 This
eect appears whether price-changes are measured between the respective last prices at which successive
aggressive orders execute (correcting for bid-ask bounce), or between the respective rst prices at which they execute, so the positive relationship between executed quantity and subsequent favorable price movements is not simply an artifact of large orders that eat through one or more levels of the orderbook.
26
Table 2: Performance of A-HFTs' Aggressive Orders (Dollars per Contract) Below Cuto Cuto
95%
CI
Above Cuto
95%
AOs Below Cuto
AOs Below Cuto
% of All AOs
% of Aggr. Volume
CI
1
(3.78, 3.89)
(7.59, 7.74)
24.31%
0.40%
5
(4.17, 4.29)
(7.62, 7.78)
43.74%
1.44%
10
(3.42, 3.55)
(7.71, 7.85)
54.64%
3.09%
15
(3.79, 3.92)
(7.71, 7.86)
56.75%
3.54%
20
(4.08, 4.20)
(7.75, 7.90)
60.82%
4.80%
As shown in Table 2, small aggressive orders represent a substantial fraction of the aggressive orders that A-HFTs place. A given A-HFT places an aggressive order of size
34
20
or less roughly once every
seconds on average, and this average interval drops to about 3 seconds during periods of intense
market activity. These small orders make up very little of the A-HFTs' total aggressive volume, but the A-HFTs' losses on them are non-trivial. On average, each A-HFT loses around day ($1.8 million, annualized) on aggressive orders of size
7.2%
$7, 150 per trading
20 or less; this loss represents approximately
of an average A-HFT's daily pro ts.
There are several reasons why a trader who consistently manages to pro t on large aggressive orders might nonetheless place unpro table small aggressive orders, including controlling risk and testing out new strategies. The A-HFTs' diering performance on small and large aggressive orders is also consistent with the pattern that we would expect to see if the small orders were generating valuable information that enabled the A-HFTs to earn greater pro ts from their large orders, as discussed in section 2.
Moreover, the A-HFTs' small aggressive orders could serve several roles at
once, so treating them as exploratory would not rule out additional reasons why they might have been placed. Although the A-HFTs' qualitative pattern of losses on small aggressive orders and morethan-osetting gains on larger aggressive orders does not necessarily imply that the A-HFTs engage in exploratory trading, it does suggest that the small orders are at least reasonable candidates for exploratory orders.
Identifying candidate exploratory orders provides the starting point for direct
tests of the exploratory trading model's sharper empirical predictions.
27
5 Testing the Exploratory Trading Hypothesis If the A-HFTs indeed engage in exploration using their small aggressive orders, the exploratory trading model generates the testable predictions presented in section 2.5, which I restate below for convenience: Relative to a benchmark that incorporates the public information about the market response following small aggressive orders placed by anyone, information from the market response following an A-HFT's small aggressive orders:
Predict.1. Explains a significant additional component of that A-HFT's earnings on subsequent aggressive orders, but
Predict.2. Does not explain any additional component of other traders' earnings on subsequent aggressive orders, and
Predict.3. Further explains by a significant margin the incidence of that A-HFT's subsequent large aggressive orders
In this section, I consider a simple numeric characterization of the market response following an aggressive order, and I directly test whether the above predictions hold.
The rst two predictions
concern the explanatory power of market-response information for the performance of subsequent aggressive orders, and I test these two predictions in the same empirical framework.
The third
prediction, concerning the incidence of large aggressive orders, requires a slightly dierent empirical approach, so I consider this prediction separately. I estimate results for the A-HFTs individually, but for compliance with con dentiality protocols, I present cross-sectional averages of these estimates. 7
Empirically, these average results are representative of the results for individual A-HFTs.
5.0.1
A Simple Measure of Market Response
De ne an aggressive order to be small if that order's submitted size is less than or equal to a speci ed size parameter, which I will denote by
q¯.
I characterize the market response to a small aggressive order using subsequent changes in orderbook depth. I examine the interval starting immediately after the arrival of a given small aggressive order and ending immediately before the arrival of the next aggressive order (which may or may not
7 Throughout
the E-mini market, there exist assorted linkages between various trading accounts (as, for example, in
the simple case where single rm trades with multiple accounts), so the trading-account divisions do not necessarily deliver appropriate atomic A-HFT units. Though the speci cs are con dential, the appropriate partition of the A-HFTs is entirely obvious. For brevity, I use individual A-HFT as shorthand to individual atomic A-HFT unit, as applicable.
28
be small), and I sum the changes in depth at the best bid and best ask that occur during this interval. As in section 4.3, I treat sell depth as negative and buy depth as positive. I also normalize these depth changes by the sign of the preceding small aggressive order to standardize across buy orders and sell orders. To simplify the analysis and stack the odds against nding signi cant results, I initially focus only on the signs of the direction-normalized depth changes. These signs merely indicate whether or not resting depth moved further in the direction of the preceding small aggressive order. For a given value of
q¯,
I construct the indicator variable
Ωk
Ω,
=
with
k th
element
Ωk
de ned by
1 if DC (k; any, q¯) > 0
(15)
0 otherwise where
DC (k; any, q¯) denotes the direction-normalized depth change following the last small aggressive
order (submitted by anyone) that arrived before the indicator variable
ΩA ,
with
k th
element
ΩA k =
ΩA k
k th
aggressive order. Similarly, I construct the
de ned by
1 if DC (k; AHF T, q¯) > 0
(16)
0 otherwise where
DC (k; AHF T, q¯)
gressive order
5.1 5.1.1
denotes the direction-normalized depth change following the last small ag-
submitted by a speci ed A-HFT
that arrived before the
k th
aggressive order.
Performance of Aggressive Orders Empirical Strategy: Overview
Though the implementation is slightly involved, my basic empirical strategy is straight-forward. First, I augment the benchmark regression from section 4.3 using either:
1. Market response information from the last small aggressive order placed by anyonei.e.,
Ω,
2. Both market response information from the last small aggressive order placed by anyone,
or
and
market response information from the last small aggressive order placed by a speci ed AHFTi.e., both
Ω and ΩA
29
After estimating both of the speci cations above, I nd the additional component of performance on larger aggressive orders explained by the second one relative to the rst. following an arbitrary small aggressive order is publicly observable.
The market response
However, because the E-mini
market operates anonymously, the distinction between a small aggressive order placed by a particular A-HFT and an arbitrary small aggressive order is private information, available only to the A-HFT who placed the order. Because the market response information from the last small aggressive order placed by anyone is weakly more recent than the market response information from last small aggressive order placed by the A-HFT, comparing the second speci cation above to the rst helps to isolate the eects attributable to private information from eects attributable to public information. Finally, I compare the additional explained performance for the speci ed A-HFT to the additional explained performance for all other traders.
Intuitively, we want to verify that the A-HFT's ex-
ploratory information provides extra explanatory power for the subsequent performance of the trader privy to that information (the A-HFT), but not for the performance of traders who aren't privy to it (everyone else). Note that everyone else includes the A-HFTs other than the speci ed A-HFT. Some A-HFT accounts and B-HFT/non-HFT accounts belong to the same rms, and various B-HFTs/nonHFTs may be either directly informed or able to make educated inferences about what one or more A-HFTs do. As a result, we should not necessarily expect exploratory information generated by an A-HFT's small orders to provide no explanatory power whatsoever for all other traders' performance. However, we should still expect the additional explanatory power for the A-HFT's performance to signi cantly exceed that for the other traders' performance.
Controlling for Public Information
Comparing the second speci cation to the rst one controls
for the eects of most public information, but there could conceivably be some public information that is correlated with the market response to a speci ed A-HFT's small aggressive orders and yet uncorrelated with the market response to small aggressive orders placed by everyone else. A natural way to handle this concern is to compare the additional explained performance for the speci ed A-HFT to the additional explained performance for some other traders who use the same public information. Although trading objectives and sophistication vary widely across many participants in the E-mini market, all of the HFTs are sophisticated, pro table traders, with similar (very short) investment horizons, so it is extremely plausible that they all use very similar public data.
Comparing the
additional explained performance for the speci ed A-HFT to that for the other HFTs therefore serves
30
as an added control for any lingering eects from public information.
5.1.2
Estimation Procedure
In the model of exploratory trading presented earlier, exploratory information was valuable only in conjunction with information about future aggressive order ow. Following this notion, I incorporate market-response information by using the indicators
Ω
and
ΩA
to partition the benchmark regression
from section 4.3. Recall that in section 4.3, I estimated equation (13),
yk
where
yk
vector
zk−1
= zk−1 Γ + k
denoted the cumulative price-change between the aggressive orders consisted of changes in resting depth between aggressive orders
the signs and signed executed quantities of aggressive orders
Ω,
k−1
through
k
and
k−1
k − 4.
k + 50,
and
k,
and the
as well as
Using the indicator
I now partition the equation above into two pieces and estimate the equation
yk = Ωk zk−1 Γa + (1 − Ωk ) zk−1 Γb + k
Next, I use the indicator
ΩA
yk
The variables
yk
and
zk−1
(17)
to further partition (17), and I estimate the equation
c d ΩA + k (k) Ωk zk−1 Γ + (1 − Ωk ) zk−1 Γ 1 − ΩA Ωk zk−1 Γe + (1 − Ωk ) zk−1 Γf + k k
=
denote the same quantities as before, and the
Γj
(18)
terms each represent
vectors of 14 coe cients. I estimate (17) and (18) for
q¯ = 1, 5, 10, 15, 20,
and for each speci cation I calculate the relative
excess performance of the speci ed A-HFT, and of all other trading accounts, on aggressive orders of size strictly greater than
q¯.
As in section 4.3, I compute the performance of aggressive order
excess of that explained by each regression by normalizing the the sign of the
k th
k th
k
in
residual from the regression by
aggressive order. I now also control for order-size eects directly by regressing the
direction-normalized residuals (for the orders of size strictly greater than
q¯) on the (unsigned) executed
quantities and a constant, then subtracting o the executed quantity multiplied by its estimated
31
regression coe cient. Controlling for size eects in this manner makes results more comparable for dierent choices of
q¯.
Size eects can be addressed by other means with negligible impact on the nal
results. For each aggressive order larger than
q¯ placed
by the A-HFT under consideration, I compute the
additional component of performance explained by (18) relative to (17) by subtracting the order's excess performance over (18) from its excess performance over (17); I stack these additional explained components in a vector that I denote by for everyone else,
ΞA .
I repeat this procedure to obtain the analogous vector
Ξee .
Equation (18) has more free parameters than (17), so
ΞA
and
Ξee
will both have positive means.
However, additional explanatory power of (18) due exclusively to the extra degrees of freedom will, in expectation, manifest equally for all traders, so the extra degrees of freedom alone should not cause
ΞA
and
5.1.3
Ξee
to dier signi cantly.
Results
I initially evaluate the rst two empirical predictions of the exploratory trading model by comparing the additional explained component of performance for each A-HFT to the additional explained component of performance for all other traders. Figure 3 displays the cross-sectional means of for dierent values of
q¯.
and
M ean (ΞA ) − M ean (Ξee ),
95%
bootstrap con dence intervals for the dierence of the pooled
displayed in Figure 4. Table 5 in Appendix D reports the numeric
Figure 3. Additional Performance Explained (95% Confidence Intervals)
values from Figures 3 and 4.
Cents per Contract
A-HFTs
Everyone Else
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
Ξee
To formally compare the gain in explanatory power for the A-HFTs to the
gain for everyone else, I construct means
ΞA
5
10
15
q-Bar
Figure 3: Additional Performance Explained (95% Con dence Intervals)
32
20
Cents per Contract
Figure 4. [A-HFT Addt'l Explained] - [Everyone Else Addt'l Explained] (95% Conf. Intervals) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
5
10
15
20
q-Bar
Figure 4: [A-HFT Addt'l Explained] - [Everyone Else Addt'l Explained] (95% CIs)
Both of the tested predictions are borne out in these results. Information about the market activity immediately following an A-HFT's smallest aggressive orders (in the form of
ΩA ) improves our ability
to explain that A-HFT's performance on larger subsequent aggressive orders by a highly signi cant margin, relative to using only information about the activity following any small aggressive order (in the form of
Ω).
Furthermore, the extra component of A-HFTs' performance on large aggressive orders
explained by using
ΩA
in addition to
Ω
is signi cantly greater than the extra component explained
for other traders. I re ne my empirical evaluation of the rst two predictions by comparing the additional explained component of performance for each A-HFT to the additional explained component of performance for the other HFTs. Consistent with the notion that certain B-HFTs may know something about what various A-HFTs are doing, the extra component of performance explained by using
Ω
ΩA
in addition to
is larger for the complementary set of HFTs than it is for the broader everyone except the A-HFT
of interest group. Nevertheless, aside from the case of
q¯ = 1,
the average added explanatory power
for each A-HFT is still signi cantly greater than is that for the complementary set of HFTs, as shown in Figure 5. (See Table 5 in Appendix D for numeric values.)
33
Cents per Contract
Figure 5. [A-HFT Addt'l Explained] - [Other HFTs Addt'l Explained] (95% Conf. Intervals) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1
0
5
10
15
20
q-Bar
Figure 5: [A-HFT Addt'l Explained] - [Other HFTs Addt'l Explained] (95% Conf. Intervals)
5.2
Incidence of A-HFTs' Larger Aggressive Orders
In this subsection I test the exploratory trading model's third prediction, namely that the market response to a given A-HFT's small aggressive order provides signi cant explanatory power for the
incidence
of that A-HFT's subsequent large aggressive orders, above and beyond that explained using
the market response to the last small aggressive order placed by anyone. Since elements of the binary
Ω-operators
correspond almost directly to the binary liquidity-state
Λ
in the exploratory trading
model, the incidence prediction can be made even more precise. In particular, all else being equal, the exploratory trading model predicts that an A-HFT will have a greater tendency to place large aggressive orders when
5.2.1
ΩA = 1
than when
ΩA = 0.
Empirical Implementation
Much as the HFT in the model from section 2 considered the signal of future aggressive order- ow as well as the liquidity state, A-HFTs consider public market data as well as exploratory information to decide when to place large aggressive orders. The size and direction of A-HFTs' aggressive orders depend on the same variables that forecast price movements, or equivalently on the forecasts of price movements themselves. On average, the signed quantity of an A-HFT's aggressive order should be an increasing function of the future price-change expected on the basis of public information. In this context, the exploratory trading model predicts that the expected future price-change will have a larger eect on the signed quantity of an A-HFT's aggressive orders when
ΩA = 1
than it will when
ΩA = 0. To test the exploratory trading model's prediction about the incidence of A-HFTs' larger aggressive
34
orders, I regress the signed quantities of a given A-HFT's aggressive orders on the associated tted values of
y
q¯,
value of
from equation (17), partitioned by
ΩA .
In other words, for a speci ed A-HFT and a given
I estimate the equation
q k = β 0 1 − ΩA ˆk + β1 ΩA ˆk + k k y ky
where
qk
denotes the signed submitted quantity of the A-HFT's
relevant tted value of
yk
function. I restrict the
β
yˆk
k th
(19)
aggressive order,
from the public-information regression (17), and
5.2.2
β0
and
β1
denotes the
is the usual indicator
coe cients to be the same across all A-HFTs. Note that the tted value
includes the public market-response information through the inclusion of
between
ΩA
yˆk
do not arise from any public information in
Ωk
in (17), so dierences
ΩA .
Results
Table 3 displays the coe cient estimates from (19) for various values of hypothesis
β 0 = β1
at the
10−15
level for all values of
q¯.
q¯.
A Wald test rejects the null
As the exploratory trading model predicts,
holding xed the price-change expected on the basis of public information, the average A-HFT places signi cantly larger aggressive orders when
ΩA = 1
than when
ΩA = 0.
Table 3: Dierential Eects of Predicted Price-Changes on A-HFT Signed Order Size
q¯ = 1
q¯ = 5
q¯ = 10
q¯ = 15
q¯ = 20
β0
13.35
13.41
13.42
13.34
13.23
(β0 Standard Error)
(0.094)
(0.093)
(0.095)
(0.095)
(0.094)
β1
15.26
15.11
14.97
15.10
15.30
(β1 Standard Error)
(0.162)
(0.169)
(0.160)
(0.159)
(0.160)
6 Practical Signi cance of Exploratory Information The empirical evidence in section 5 provides strong support for the hypothesis that the A-HFTs engage in exploratory trading as modeled in section 2. However, while these results indicate that exploratory trading plays some part in what the A-HFTs are doing to pro t from their aggressive orders, the
35
results tell us little about how large that part is. Estimates of the additional component of the A-HFTs' aggressive-order earnings directly explained by the private information in
ΩA
are likely to dramatically understate the true contribution of ex-
ploratory information, for two reasons. First, exploratory information.
ΩA
is nearly the simplest possible characterization of
Representations of exploratory information richer than
ΩA
are extremely
easy to construct. For example, an obvious extension would be to consider the not only the sign, but also the
magnitude
of the direction-normalized depth change following an exploratory order. Regard-
less of the particular representation of exploratory information used, though, the additional explained component of A-HFTs' pro ts on the aggressive orders they place is likely to understate the true gains from exploration.
As the simple model in section 2 illustrates, exploratory information is valuable
in large part because it enables a trader to avoid placing unpro table aggressive orders.
However,
estimates of the additional explained component of pro ts on A-HFTs' aggressive orders necessarily omit the eects of such avoided losses. While this bias, if anything, makes the preceding ndings of statistical signi cance all the more compelling, it also complicates the task of properly determining the practical importance of exploratory information.
6.1
Simulated Trading Strategies
To investigate the gains from exploratory information, including the gains from avoiding unpro table aggressive orders, I examine the eects of incorporating market-response information from small aggressive orders into simulated trading strategies. The key advantage of working with these simulated trading strategies is that avoided unpro table aggressive orders can be observed directly. The basic trading strategy that I consider is a simple adaptation of the benchmark regression from section 4.3. I specify a threshold value, and the strategy entails nothing more than placing an aggressive order with the same sign as
yˆk
whenever
|yˆk | exceeds that threshold.
feasible (in the sense of using only information available before time compute the forecast of the future price movement,
To make this strategy
t to determine the time-t action) I
yˆk , using the regression coe cients estimated from
the previous day's data. I incorporate market-response information into this strategy by modifying the rule for placing aggressive orders to, place an aggressive order (with the same sign as only if all three of the following conditions hold:
• |yˆk |
exceeds its speci ed threshold,
36
yˆk )
if and
•
The direction-normalized depth-change following the last small aggressive order (placed by anyone) exceeds a speci ed threshold, and
•
The direction-normalized depth-change following the last small aggressive order
A-HFT
placed by an
exceeds a (possibly dierent) speci ed threshold.
Choosing a threshold of
−∞
will eectively remove any of these conditions.
Each strategy yields a set of times to place aggressive orders, and the associated direction for each order.
To measure the performance of a given strategy, I compute the average pro tability of the
indicated orders in the usual manner, with the assumption that these aggressive orders are all of a uniform size. Relative to A-HFTs' losses on small aggressive orders, the additional component of A-HFTs' pro ts directly explained using
ΩA
is smallest when
q¯ = 10,
and I present results for
q¯ = 10
to highlight
the impact of accounting for avoided losses on estimates of the gains from exploratory information. Results for other values of
6.2
q¯ are
similar.
Three Speci c Strategies
All three threshold parameters aect strategy performance, so to emphasize the role of market-response information, I present results with the threshold for does not alter the qualitative results.
|yˆk |
held xed.
Varying the threshold for
|ˆ yk |
In particular, it is not possible to achieve the same gains in
performance that result from incorporating exploratory information by merely raising the threshold for
|ˆ yk |.
The forecast
yˆk
uses coe cients estimated from the previous day's data, and these forecasts
exhibit increasing bias as the
zk−1
observations assume more extreme values.
I consider a range of threshold values for the direction-normalized depth-change following the last small aggressive order placed by anyone, but, for expositional clarity, I present results for three illustrative threshold choices for the direction-normalized depth-change following the last small aggressive order placed by an A-HFT. Speci cally, I consider thresholds of information),
0
(the same information contained in
ΩA ),
and
417
−∞ (the
(no A-HFT market-response
99th
percentile value). Figure
6 displays the performance of these three strategies over a range of threshold values for the market response following arbitrary small aggressive orders.
37
Avg. Gross Earnings per Contract (Dollars)
Figure 6. Absolute Gains from Exploratory Information No A-HFT Info
A-HFT DC>0
A-HFT DC>99th %tile
8.25 8.00 7.75 7.50 7.25 7.00 6.75 6.50 6.25 6.00 50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
Percentile Cutoff for Depth-Change Following Last Small Aggressive Order by Anyone
Figure 6: Absolute Gains from Exploratory Information
While the performance gains from incorporating A-HFT exploratory information are obvious, an equally important feature of the results above is more subtle. The A-HFTs' average gross earnings on aggressive orders over size 10 of
$7.78
per contract are well above the peak performance of the
strategy that uses only public information, but substantially below the performance of the strategy that incorporates the A-HFTs' exploratory information with the higher threshold. This is exactly the pattern that we should expect, given that the former strategy excludes information that is available to the A-HFTs and the latter strategy includes information that is not available to any individual A-HFT, so these results help to con rm the relevance and validity of this simulation methodology.
6.2.1
Gains from Exploration Relative to Losses on Exploratory Orders
Although the two strategies that incorporate exploratory information from the A-HFTs' small aggressive orders outperform the strategy that does not, the orders that generated the exploratory information were costly. To compare the gains from this exploratory information to the costs of acquiring it, I rst multiply the increases in per-contract earnings for the two exploratory strategies (scaled by the respective number of orders relative to the public-information strategy) by the A-HFTs' combined aggressive volume on orders over size 10.
8
I then divide these calibrated gains by the A-HFTs' actual
losses on aggressive orders size 10 and under. Figure 7 displays the calibrated ratio of additional gains to losses for each exploratory simulated
8 The
two strategies that incorporate exploratory information select subsets of the aggressive order placement times
generated by the public-information-only strategy. Although the selected orders tend to be more pro table, they are also fewer in number.
38
strategy over a range of threshold values for the market response following arbitrary small aggressive orders. Using information from the A-HFTs' exploratory orders analogous to that in gains are roughly
15%
ΩA , the additional
larger than the losses on exploratory orders. Whereas the extra component
of the A-HFTs' performance directly explained using
ΩA
represents less than
5%
of A-HFTs' losses
on exploratory orders, the analogous estimated performance increases more than oset the costs of exploration once we include the gains from avoiding unpro table aggressive orders. In the case of the strategy that employs information from the A-HFTs' exploratory orders with the higher threshold, the estimated gains from exploration exceed the costs by more than one-third.
Figure 7. Gains from A-HFT Exploratory Info Relative to Losses on Exploratory Orders
Ratio of Extra Gains to Losses
A-HFT DC>0
A-HFT DC>99th %tile
1.6 1.5 1.4 1.3 1.2 1.1 1.0 50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
Percentile Cutoff for Depth-Change Following Last Small Aggressive Order by Anyone
Figure 7: Gains from A-HFT Exploratory Info Relative to Losses on Exploratory Orders
Even after netting out the calibrated losses on exploratory orders from the better-performing exploratory-information simulated strategy in 6.2, the simulated performance exceeds the maximum pro tability hurdle among HFTs of
$6.41
per aggressively traded contract. An almost trivial trading
strategy that incorporates exploratory trading appears to be pro table, suggesting very strongly that exploratory trading is at least su cient to explain how a trader in the E-mini market could consistently pro t on average from her aggressive orders.
7 Discussion 7.1
Broader Opportunities for Exploratory Gains from Aggressive Orders
The empirical analysis in the preceding sections focused on the information generated by the A-HFTs' smallest aggressive orders. While the otherwise-perplexing unpro tability of these orders made them
39
the most obvious starting point for an empirical study of exploratory trading, there is no theoretical reason why these small orders should be the sole source of exploratory information. In the baseline exploratory trading model, it was only to highlight the key aspects of the model that I assumed the HFT's period-1 order was expected to lose money and served no purpose other than exploration. In principle, even aggressive orders that an A-HFT expects to be directly pro table could produce valuable, private, exploratory information. section 5.1 setting
To investigate this possibility, I repeat the analysis of
q¯ = 25, 30, 35, 40, 45, 50, 60, 75, 90.
included with each increase of
q¯ beyond q¯ = 20
The A-HFTs' incremental aggressive orders
are directly pro table on average, and yet the market
response following these orders still provides signi cantly more additional explanatory power for the A-HFTs' performance on larger aggressive orders than it provides for that of other traders. Indeed, the additional explained components of the A-HFTs' performance are markedly larger than those for
q¯ = 1, 5, ..., 20Figure ; see Figure 8, Addt'l and see Table 5- in Appendix for Explained] numeric results. 8. [A-HFT Explained] [Everyone Else D Addt'l (95% Conf. Bands)
1.2
Cents per Contract
1 0.8 0.6 0.4 0.2 0 0
10
20
30
40
50
60
70
80
90
100
q-Bar
Figure 8: [A-HFT Addt'l Explained] - [Everyone Else Addt'l Explained] (95% CIs)
These results have the interesting implication that the A-HFTs enjoy natural and almost inevitable economies of scalesimply by being in the market and engaging in lots of aggressive trading, they automatically generate lots of valuable, private information. Other, more obvious economies of scale and scope likely exist for high-frequency traders (e.g., tiered trading costs, applicability of similar algorithms across dierent markets), but the economies of scale arising from exploratory information appear to be new. The impressive performance of the extremely simple simulated strategies in section
40
6 casts doubt on the standard fallback of intellectual capital as a barrier to entry.
Although the
A-HFTs earn positive pro ts on average, their marginal pro ts need not be strictly positive, so there may be no incentive for new A-HFTs to enter. However, should the structure of the A-HFT industry indicate the existence of some barriers to entry, the A-HFTs' apparent economies of scale could potentially act as one such barrier.
Industrial organization of high-frequency trading entities is an
intriguing open area for future investigation, but detailed treatment lies beyond the scope of this paper.
7.2
Exploratory Trading and Speed
Evidence in this paper provides empirical justi cation for using the exploratory trading model to draw conclusions about real-world high-frequency trading. Further analysis of the exploratory trading model reveals natural connections between exploration and two important concepts of speed. These connections in turn help to illuminate the role that the two types of speed play in high-frequency trading.
7.2.1
Low Latency
One common measure of trading speed is
latency the amount of time required for messages to pass
back and forth between a trader and the market.
While low-latency operation and high-frequency
trading are not equivalent, minimal latency is nonetheless a hallmark of high-frequency traders. HFTs can certainly react and communicate faster than some other market participants, but analogous differences in the
relative
reaction speed of various traders long predate high-frequency trading. For a
trader who can identify pro table trading opportunities, there is obvious value to possessing latency low enough to take advantage of these opportunities before they disappear.
The new insight from
the exploratory trading model concerns the more subtle matter of how low latency connects to the identi cation of such opportunities, that is, why it might matter for latency to be low in absolute terms. In the model of exploratory trading developed in section 2, the HFT's inference about
Λ
on the
basis of market activity following his aggressive order in period 1 implicitly depends on a notion related to latency. If we suppose that random noise perturbs the orderbook, say according to a Poisson arrival process, then the amount of noise present in the HFT's observation of the market response in some interval following his aggressive order will depend on the duration of that interval. The duration of this
41
interval will depend in large part upon the rate at which market data is collected and disseminated to the HFT, that is, the temporal resolution of the HFT's data. Although this temporal resolution does not directly depend on the HFT's latency, the temporal resolution of the HFT's market information
does
implicitly constrain how quickly the HFT can learn about market events.
The ner temporal resolution required for low-latency operation enables low-latency traders to obtain meaningfuland empirically valuableinformation about the market activity immediately following their aggressive orders, and this information degrades at coarser temporal resolutions. The empirical results from section 5.1.3 provide a concrete illustration of this eect.
The changes in
resting depth immediately following an arbitrary aggressive order are less useful for forecasting price movements than are the analogous changes following an A-HFT's aggressive order, but the two can only be distinguished (by the A-HFT) in data with a su cient level of temporal disaggregation.
7.2.2
High Frequency
Exploratory trading bears a natural relationship to the practice of placing large numbers of aggressive orderswhat might be considered high-frequency trading in the most literal sense. Exploratory information generated by a given aggressive order is only valuable to the extent that it can be used to improve subsequent trading performance. Because exploratory information remains relevant for only some nite period, the value of exploratory information diminishes as the average interval between a trader's orders lengthens.
The exploratory trading model readily captures this
eect if we relax the simplifying assumption that the liquidity state periods 1 and 2. Suppose that
τ,
a second
1 − τ,
Λ
Λ
Λ
remains the same between
evolves according to a Markov process, such that with probability
is drawn in period 2 (from the same distribution as in period 1), and with probability
the original value from period 1 persists in period 2.
of period 1, and this length increases from zero to in nity as
Intuitively,
τ
τ
parametrizes the length
increases from zero to unity. As
τ
tends towards unityi.e., as the length of period 1 increases to in nitythe liquidity state in period 1 becomes progressively less informative about the liquidity state in period 2. As discussed in section 7.1, both theory and empirical evidence suggest that almost any aggressive order that a trader places generates some amount of exploratory information.
Consequently, as a
trader places aggressive orders in greater numbers, he will gain access to greater amounts of exploratory information. Furthermore, the average time interval between a trader's aggressive orders necessarily shrinks as the number of those orders grows, so the exploratory information produced by each order
42
tends to become more valuable to the trader.
These synergistic eects dramatically magnify the
potential gains from exploratory information for traders who place large numbers of aggressive orders.
7.2.3
Latency Détente
There has been much speculation about HFTs engaging in an arms race for ever-faster processing and ever-lower latency.
If high-frequency trading entailed nothing more than reacting to publicly
observable trading opportunities before anyone else, HFTs would indeed face nearly unbounded incentives to be faster than their competitors. While reaction speed is certainly one dimension along which HFTs compete, the empirical evidence of exploratory trading suggests that the A-HFTs, at least, can also compete along another dimensionexploration. Since exploratory trading provides the A-HFTs with private information, a trader who uses only public information will not necessarily be able to dominate the A-HFTs, even if that trader is faster than every A-HFT. Similarly, an A-HFT could potentially compensate for having (slightly) slower reactions than the other A-HFTs by engaging in greater levels of exploration. Because A-HFTs can compete on their levels of exploration as well as on speed, even in a longrun equilibrium, technology suppliers will not necessarily capture all economic rents that HFTs earn. Provided that exploratory orders are costly on average, A-HFTs' losses on those exploratory orders are transfers to the A-HFTs' respective liquidity-providing counterparties.
7.3
Beyond A-HFTs: Other HFTs and Other Markets
Exploratory trading is not universally relevant to all HFT activity in all markets. However, exploratory trading in the E-mini market depends only on the market's structure and aggregate dynamicsit does not depend directly on any speci c features of the E-mini contract. The prevalence of exploratory trading in other markets is ultimately an empirical matter, but markets similar to the E-mini in size and structure could easily support exploratory trading. Even in the E-mini market, an important component of HFT activity lies outside the immediate province of the exploratory trading model. Nevertheless, the scope for exploratory trading extends well beyond the aggressive activity of A-HFTs considered thus far. Though I have focused on the AHFTs up to this point, the B-HFTs could also reap exploratory rewards from their aggressive orders, as could potentially any trader with similar capabilities. The B-HFTs' overall performance on aggressive orders does not present the same ostensible aront to market e ciency as does that of the A-HFTs,
43
but the B-HFTs' aggressive orders nonetheless outperform both those of non-HFTs, and the baseline econometric benchmark, by a wide margin. If inventory management or risk-control considerations force B-HFTs to place unpro table aggressive orders, exploratory trading could help to explain how the B-HFTs mitigate the associated losses. Alternatively, if nothing forces the B-HFTs to place aggressive orders, then the B-HFTs' consistent losses from aggressive trading are puzzling in their own right, much as the A-HFTs' losses on small aggressive orders were. Although the B-HFTs do not recoup their losses on other aggressive orders as do the A-HFTs, they make enough from their passive trading to earn positive pro ts overall. Passive trading strategies, just like aggressive ones, would bene t from the superior price forecasting that exploratory information makes possible, so exploratory trading could help to explain the activity of 9
B-HFTs in this scenario as well.
8 Conclusion Empirical evidence strongly suggests that the concept of exploratory trading developed in this paper helps to explain the mechanism underlying certain HFTs' superior capacity to pro tably anticipate price movements in the E-mini market. The exploratory trading model also illuminates the manner in which these HFTs bene t from low-latency capabilities and from their submission of large numbers of aggressive orders. Exploratory trading is a form of costly information acquisition, albeit an unfamiliar one. HFTs who engage in exploratory trading are doing something more than merely reacting to public information sooner other market participants. This raises the possibility that HFTs, through exploratory trading, uniquely contribute to the process of e cient price discovery.
However, exploratory trading diers
from traditional costly information acquisition in several important respects. First, the information that exploratory trading generates does not relate directly to the traded asset's fundamental value,
,
but rather pertains to unobservable aspects of market conditions that could eventually become public
ex-post, through ordinary market interactions.
Also, because exploratory trading operates through the
market mechanism itself, exploration exerts direct eects on the market, distinct from the subsequent eects of the information that it generates. Finally, since HFTs appear to trade ahead of predictable demand innovationsalbeit in a sophisticatedly selective mannerthe research of De Long
9 Total
et al.
trading pro ts from any transaction net to zero, so if a trader earns money on an aggressive order, his passive
counter-party loses money. Since exploratory information is valuable to an aggressor, it follows immediately that it is also valuable to a passor.
44
(1990) potentially suggests that HFTs could have a destabilizing in uence on prices if suitable positivefeedback mechanisms exist. Comprehensive analysis of the theoretical and empirical aspects of these myriad issues lies beyond the scope of this paper, but the theory and evidence presented herein provide a starting point from which to rigorously address the market-quality implications of high-frequency trading going forward.
45
References [1] Matthew Baron, Jonathan Brogaard, and Andrei Kirilenko. The trading pro ts of high frequency traders. September 2012.
[2] Bruno Biais, Thierry Foucault, and Sophie Moinas. Equilibrium algorithmic trading. Working Paper, October 2010.
[3] Jonathan Brogaard, Terrence Hendershott, and Ryan Riordan. High frequency trading and price discovery. 2012.
[4] Giovanni Cespa and Thierry Foucault.
Insiders-outsiders, transparency and the value of the
ticker. Queen Mary University Dept. of Economics Working Paper No. 628, April 2008.
[5] J.Bradford de Long, Andrei Shleifer, Lawrence H. Summers, and Robert J. Waldmann. Positive feedback investment strategies and destabilizing rational speculation.
The Journal of Finance,
45(2):379395, June 1990.
[6] Lawrence R. Glosten and Paul R. Milgrom. Bid, ask and transaction prices in a specialist market with heterogeneously informed traders.
Journal of Financial Economics, 14:71100, 1985.
[7] Sanford J. Grossman and Merton H. Miller.
Liquidity and market structure.
The Journal of
Finance, 43(3), July 1988. [8] Bjorn Hagstromer and Lars Norden. The diversity of high-frequency traders. Stockholm University School of Business, May 2013.
[9] Joel Hasbrouck and Gideon Saar. Technology and liquidity provision: The blurring of traditional de nitions.
Journal of Financial Markets, 12:143172, 2009.
[10] Joel Hasbrouck and Gideon Saar.
Low-latency trading.
NYU Stern/ Cornell GSM Working
Paper, May 2011.
[11] Terrence Hendershott, Charles M. Jones, and Albert J. Menkveld. improve liquidity?
Does algorithmic trading
The Journal of Finance, 66(1), February 2011.
[12] Terrence Hendershott and Ryan Riordan. Algorithmic trading and information. NET Institute Working Paper No. 09-08, September 2009.
46
[13] Nicholas H. Hirschey. Do high-frequency traders anticipate buying and selling pressure? December 2011.
[14] Robert Jarrow and Philip Protter. A dysfunctional role of high frequency trading in electronic markets. Johnson School Research Paper Series No. 08-2011, March 2011.
[15] Andrei Kirilenko, Mehrdad Samadi, Albert S. Kyle, and Tugkan Tuzun. The ash crash: The impact of high frequency trading on an electronic market. October 2010.
[16] Albert S. Kyle. Continuous auctions and insider trading.
Econometrica, 53(6):13151335, Novem-
ber 1985.
[17] J.Chris Leach and Ananth N Madhavan. Intertemporal price discovery by market makers: Active versus passive learning.
Journal of Financial Intermediation, 2(2):Pages 207235, June 1992.
[18] J.Chris Leach and Ananth N. Madhavan. Price experimentation and security market structure.
The Review of Financial Studies, 6(2):pp. 375404, 1993. [19] U.S. SEC. Concept release on equity market structure, concept release no. 34-61358. FileNo. 17 CFR Part 242 [Release No. 34-61358; File No. S7-02-10] RIN 3235-AK47, January 2010.
[20] U.S. SEC. Findings regarding the market events of may 6, 2010, September 2010.
[21] Hans R. Stoll. Inferring the components of the bid-ask spread: Theory and empirical tests.
Journal of Finance, 44(1):115134, March 1989.
47
The
A Exploratory Trading Model Details A.1 Let
st
Solving the Baseline Exploratory Trading Model
A.1.1
If
Solving the Model: Period 2
ϕ = 0,
to set
qt .
denote the sign of
the HFT's optimal choice is to not submit an aggressive order in period 2, or equivalently,
|q2 | = 0.
If
ϕ 6= 0,
then it is optimal for the HFT to set
so we only need to determine the optimal magnitude,
|q2 | .
xed, we can restrict attention to corner solutions (0 or of generality. Note that if
q2 = 0,
Suppose that the HFT sets
then
π2 = 0,
|q2 | = N .
N)
s2 = ϕ
Because
(unless the optimal
π2
is linear in
ϕ
and
is zero),
|q2 | when s2
for the optimal choice of
regardless of the values of
|q2 |
|q2 |
is held
without loss
Λ.
Without loss of generality, assume that
s2 = ϕ 6= 0.
The
HFT's period-2 pro ts are given by
π˜2 =
N (1 − α) if Λ = U −N α
where the tilde on
π˜2
(20)
if Λ = A
denotes the fact that the HFT's choice of
q2
does not condition on the value of
Λ.
HFT Does Not Know
Λ
If the HFT does not know the value of
the HFT's expected period-2 pro t if he sets
|q2 | = N
E [π˜2 |ϕ 6= 0, |q2 | = N ]
|q2 | = N
(and
s2 = ϕ)
ϕ,
then in the case where
ϕ 6= 0,
is
= uN (1 − α) − (1 − u) N α =
Taking expectations with respect to
Λ,
(21)
(u − α) N
we nd that the
ex-ante
expectation of
π˜2
when the HFT sets
is given by
E [π˜2 | |q2 | = N ] = v (u − α) N
48
(22)
u − α < 0,
When
ex-ante
if the HFT did not know
expectation of
π˜2
Λ,
he would set
q2 = 0
rather than
|q2 | = N .
Hence the
is
E [π˜2 ] = max {v (u − α) N, 0} HFT Knows
only when
Λ
Λ=U
Next, if the HFT and
ϕ 6= 0.
does
know the value of
Λ,
(23)
then he will set
|q2 | = N
(and
Denoting the HFT's period-2 pro ts from this strategy by
E [πˆ2 |ϕ 6= 0]
E [πˆ2 ]
πˆ2 ,
s2 = ϕ) we nd
= u (1 − α) N
(24)
=
(u − α) N + α (1 − u) N
=
vu (1 − α) N
=
v (u − α) N + vα (1 − u) N
(25)
Note that
E [πˆ2 ] > max {v (u − α) N, 0}
(26)
so the HFT's expected period-2 pro ts are strictly greater when he knows know
Λ
than when he doesn't
Λ.
A.1.2
Solving the Model: Period 1
At the start of period 1, the HFT knows neither
ϕ nor Λ, but he faces the same trading costs, α, as he
does in period 2. Consequently, the HFT's expected direct trading pro ts from a period-1 aggressive order are negative:
E [π1 |q1 ]
=
E [|q1 | (s1 y − α) |s1 , q1 ]
=
|q1 | s1 E [y] − α |q1 |
=
−α |q1 |
The second equality relies on the assumptions that
q1 ,
while the nal equality uses the fact that
ϕ
and
Λ
(and hence
yields no more information about
y)
are independent of
s1
and
E [y] = 0.
Since there is no noise in this baseline model, the HFT learns order that he places in the rst period with
(27)
|q1 | ≥ 1.
Λ
perfectly from any aggressive
An aggressive order of size greater than one
Λ than a one-contract aggressive order in this setting, but the larger
49
aggressive order incurs additional expected losses. attention to the case of If the HFT sets
q1 = 0
q1 = 0 ,
and the case of
he neither learns
Thus without loss of generality, we can restrict
|q1 | = 1.
Λ
nor incurs any direct losses in period 1, so his total
expected pro ts are simply
E [πtotal |q1 = 0]
= E [π˜2 ] max {v (u − α) N, 0}
=
Alternatively, if the HFT sets
|q1 | = 1,
(28)
his total expected pro ts are given by
E [πtotal | |q1 | = 1]
= −α |q1 | + E [πˆ2 ]
(29)
= vu (1 − α) N − α
A.1.3
Comparative Statics for Model Parameters
Recall that when the exogenous aggressive order- ow is described by
ϕ = 0, the HFT does not have any
pro table period-2 trading opportunities in either liquidity state. The probability that the parameter
ϕ 6= 0, given by
v , represents the extent to which the exogenous aggressive order- ow is predictable.
To
characterize how various parameters aect the viability of exploratory trading, I consider the minimal value of
v for which the HFT nds it optimal to engage in period-1 (i.e., exploratory) trading.
this minimal value by
v,
we have
v= The closer is
∂v ∂α
> 0,
∂v ∂N
v
α, the HFT will take the same action in period 2 when he knows that Λ = U as when he doesn't know Λ, so knowledge of the liquidity state is less likely to change the HFT's period-2 actions when u is large. In the case of 1 u > α, equation (30) becomes v = (1−u)N , and exploratory trading indeed becomes less viable as u approaches 1.
50
A.1.4
Remark on the Sequence of Events
The central results of the model would not change if the HFT observed the signal of future aggressive order- ow
before
deciding whether to engage in exploratory trading, rather than observing it after
deciding. However, the sequence of events outlined in section 2, in which the HFT must choose whether or not to explore before he observes
ϕ,
is more appropriate from an empirical perspective. For the
HFT to learn about the liquidity state after he submits an aggressive order, he must wait for 1) his order to reach the market and execute, 2) information about that execution to reach other traders, 3) other traders to decide what to do, 4) other traders' decisions to reach the market, and 5) information about the market response to get back to him.
Of these ve steps, (1), (2), (4) and (5) each take
approximately
3−4 milliseconds, and (3) takes considerably longer, perhaps 3−20 milliseconds, for an
overall total of
15−40 milliseconds.
An HFT who has already done his exploration will be able to take
advantage of predictable aggressive order- ow long before an HFT who only engages in exploratory trading after seeing an order- ow signal.
A.2
Solving the Model of Section 2.4
If the HFT places an order in the rst period, it follows immediately from the baseline model results that his expected total pro ts are given by
E [πtotal | |q1 | = 1]
= N vu (1 − α) − α
(31)
However, the HFT's expected pro ts if he does not place an order in period 1 are higher than in the baseline model, because the HFT now learns something from the depth changes following the other trader's aggressive order. If resting depth weakly replenishes after that order, the HFT learns with certainty that the liquidity state is accommodating (i.e.,
Λ = A),
so the HFT will not submit an
aggressive order in period 2, and his total pro ts will be zero. Alternatively, if resting depth further depletes following the aggressive order in period 1 (denote this event by
51
g1 ),
we have
P {Λ = U |g1 }
= = = =
P {Λ = U, and g1 } P {g1 } P {g1 |Λ = U } P {Λ = U } P {g1 |Λ = U } P {Λ = U } + P {g1 |Λ = A} P {Λ = A} 1 ∗ P {Λ = U } 1 ∗ P {Λ = U } + ρ ∗ P {Λ = A} u u + ρ (1 − u)
(32)
(33)
It follows immediately from the analogous result in the baseline model that the HFT's expected period-2 pro ts are given by
E [π2 |AO by someone else] = max N v
u − α ,0 u + ρ (1 − u)
(34)
Since the HFT does not place an order in the rst period, his expected total pro ts equal his expected period-2 pro ts.
Overall, then, the HFT's expected total pro t if he observes an aggressive order
placed by someone else in period 1 but does not place an aggressive order himself, is
E [πtotal |AO by someone else] = max N v
52
u − α ,0 u + ρ (1 − u)
(35)
B Measuring Aggressive Orders' Pro tability Calculating round-trip pro ts using a FIFO or LIFO approach is not a useful way to measure the pro tability of individual aggressive orders. Even the most aggressive HFTs engage in some passive trading, so a FIFO/LIFO-round-trip measure would either confound aggressive trades with passive trades, or require some arbitrary assumption to distinguish between inventory acquired passively and inventory acquired aggressively (on top of the already-arbitrary assumption of FIFO or LIFO). A second, more general problem is that a measurement scheme based on inventory round-trips will always combine at least two orders (an entry and an exit), so such measurement schemes do not actually measure the pro tability of
individual
aggressive orders.
In this appendix, I provide rigorous justi cation for the claim that the average expected pro t from an aggressive order in the E-mini market equals the expected favorable price movement, minus trading/clearing fees and half the bid-ask spread. After presenting the formal proof, I discuss details of empirically estimating expected favorable price movement.
B.1
Preliminaries
Trading/clearing fees apply equally to both passively and aggressively traded E-mini contracts, so to simplify the exposition, I will initially ignore these fees. Similarly, I make the simplifying assumption that the bid-ask spread is constant, and identically equal to one tick; for the E-mini market, this assumption entails minimal loss of generality. In the E-mini market, the pro tability of individual aggressive orders can be considered in isolation from passive orders. Because E-mini contracts can be created directly by buyers and sellers, a trader's net inventory position does not constrain his ability to participate in a given trade
11
.
As long as
he can nd a buyer, a trader who wishes to sell an E-mini contract can always do so, regardless of whether he has a preexisting long position. More generally, if a trader enters a position aggressively then exits it passively, he could have conducted the passive transaction even if he hadn't engaged in the preceding aggressive transaction. While a desire to dispose of passively-acquired inventory might
motivate
a trader to submit an aggressive order, the question of underlying motivation is distinct from
the question of whether the aggressive order was directly pro table.
11 The
one exception would arise in the extremely rare event that a trader who did not qualify for a position-limit
exemption held so many contracts (either long or short) that his inventory after the trade would exceed the position limit of
100, 000
E-mini contracts. For HFTs, this minor exception can safely be ignored.
53
B.2
Formal Argument
With these preliminaries established, I turn to the rigorous argument. ecutes
J.
J
aggressive sell orders of size one, and
J
Consider a trader who ex-
aggressive buy orders of size one, for some large
Following the remarks above, the trader's passive transactions can be ignored. Let the average
direction-normalized price change after these aggressive orders be does not depend on
ϑ˜ ≡ ϑ
2J 2J−1
ticks for some
ϑ that
J.
First, suppose that the trader always submits an aggressive sell after an aggressive buy, and always submits an aggressive buy after an aggressive sell. Without loss of generality, assume that the trader's rst aggressive order is a buy. The trader's combined pro t from all
π2J
2J
aggressive orders is
= −a1 + b2 − a3 + b4 − . . . − a2J−1 + b2J
(36)
= −a1 + (a2 − 1) − a3 + (a4 − 1) − . . . − a2J−1 + (a2J − 1)
(37)
=
−a1 + a2 − a3 + a4 − . . . − a2J−1 + a2J − J
(38)
= −a1 + (a1 + ζb,1 ) − (a2 + ζs,2 ) + (a3 + ζb,2 ) − . . .
(39)
. . . − (a2J−2 + ζs,J ) + (a2J−1 + ζb,J ) − J =
J X
(a2i−1 + ζb,i ) −
i=1
=
J X
ak
and
bk
order executes,
(a2j−2 + ζs,j ) − a1 − J
(40)
j=2
a2i−1 − a1 +
i=1
where
J X
J−1 X
a2j +
j=1
J X
ζb,i −
J X
ζs,j − J
(41)
j=2
i=1
respectively denote the prevailing best ask and best bid at the time the
ζb,r
denotes the change in midpoint price following the
ζs,r denotes the change in midpoint P PJ J 1 ϑ ≡ 2J (−ζ ) . ζ + b,r s,r r=1 r=1 and
price following the
Next, taking expectations, we nd
54
rth
rth
k th aggressive
aggressive buy order,
aggressive sell order.
Note that
E [π2J ]
=
J X
E [a2i−1 ] − E [a1 ] +
i=1
+
J X
J−1 X
E [a2j ]
(42)
j=1
E [ζb,i ] −
i=1
J X
E [ζs,j ] − J
j=2
h i h i = JE [a1 ] − E [a1 ] − (J − 1) E [a1 ] + JE ϑ˜ − (J − 1) −E ϑ˜ − J h i = (2J − 1) E ϑ˜ − J 2J −J = (2J − 1) E [ϑ] 2J − 1
(43) (44) (45)
= J (2E [ϑ] − 1)
(46)
where the second equality uses the assumption that midpoint prices follow a martingale with respect to their natural ltration, together with the assumption of a constant bid-ask spread.
From the
nal equality above, it follows immediately that the trader's average expected pro t on an individual aggressive order is given by
1 1 E [π2J ] = E [ϑ] − 2J 2
(47)
Finally, note that none of the calculations above relied on the assumption that the aggressive orders alternated between buys and sells (this only simpli ed the notation).
It follows immediately from
grouping together multiple aggressive orders of the same sign that the result would hold for orders of varying sizes, provided that the overall aggressive buy and aggressive sell volumes were equal. Under the usual regularity conditions, as
B.3
h i J → ∞, ϑ˜ →A.S. limJ→∞ E ϑ˜ = E [ϑ].
Obtaining Unbiased Estimates
Recall that the discussion in section 4.1 implied that we can estimate the pro tability of an HFT's aggressive order using the (direction-normalized) accumulated price-changes following that aggressive order out to some time past the HFT's maximum forecasting horizon. If we choose too short an accumulation window, the resulting estimates of the long-run direction-normalized average price changes following an HFT's aggressive orders will be biased downward. This enables us to empirically determine an adequate accumulation period by calculating cumulative direction-normalized price changes over longer and longer windows until their mean ceases to signi cantly increase Market activity varies considerably in its intensity throughout a trading day, so event-time, which
55
I measure in terms of aggressive order arrivals, provides a more uniform standard for temporal measurements than does clock-time. Empirically, an accumulation period of about 30 aggressive orders su ces to obtain unbiased estimates of the price movement following an HFT's aggressive order, but I consider results for an accumulation period of 50 aggressive orders to allow a wide margin for error. The mean direction-normalized price changes following individual HFTs' aggressive orders does not dier signi cantly for accumulation periods of 50, 200, or 500 aggressive orders, even if we distinguish aggressive orders by size. The same holds true for aggressive orders placed by non-HFTs. Using too long an accumulation period will not bias the estimates, but it will introduce unnecessary noise, so I opt for an accumulation period of 50 aggressive orders. As I discuss at greater length in section 4.3, future price movements are moderately predictable from past aggressive order ow and orderbook activity, but only at very short horizons.
Of the
variables that meaningfully forecast future price changes, the direction of aggressive order ow is by far the most persistent, but even its forecasting power diminishes to nonexistence for price movements more than either about 12 aggressive orders or 200 milliseconds in the future. The adequacy of a
30+
aggressive order accumulation period is entirely consistent with these results. As a simple empirical check on the validity of direction-normalized cumulative price changes as a proxy for the pro tability of aggressive orders, I use each HFT's explicit overall pro ts and passive trading volume, together with the pro ts on aggressive orders as measured by the proxy, to back out the HFT's implicit pro t on each passively traded contract.
The resulting estimates of HFTs'
respective pro ts from passive transactions are all plausible from a theoretical perspective, and are comparable to non-HFTs' implicit performance on passive trades.
56
C Benchmark Regression Results In this appendix, I present and discuss results from regression (13) of section 4.3. Recall that for each trading day in my sample, I regress the cumulative price-change (in dollars) between the aggressive orders
k
depth between aggressive orders
k + 50,
and
k−1
and
denoted
k
k−1
through
k − 4.
on the following variables: changes in resting
at each of the six price levels within two ticks of the best
bid or best ask, the signs of aggressive orders of aggressive orders
yk ,
k−1
through
k − 4,
and the signed executed quantities
For symmetry, I adopt the convention that sell depth is
negative, and buy depth is positive, so that an increase in buy depth has the same sign as a decrease in sell depth. I estimate the equation
yk
=
zk−1 Γ + k
(48)
:= γ1 d1k−1 + . . . + γ6 d6k−1 +
(49)
γ7 signk−1 + . . . + γ10 signk−4 + γ11 qk−1 + . . . + γ14 qk−4 + k
where
drk−1
r=4
corresponds to the best ask),
denotes the change in resting depth at price level
signl
r (r = 3
corresponds to the best bid,
denotes the sign of aggressive order
l,
and
ql
denotes the
signed executed quantity of aggressive order l. Table 4 summarizes the estimates from the regression above, computed over my entire sample. All of the variables are antisymmetrical for buys and sells, and so have means extremely close to zero, but the mean magnitudes in the rightmost column of Table 4 provide some context for scale.
57
Table 4: Estimates from Benchmark Regression Coe cient
(×1000)
Robust t-Statistic
Variable Avg. Magnitude
-0.90
-1.02
4.13
bid−1 dbest k−1
-2.08
-4.29
10.8
bid dbest k−1
1.13
4.94
23.1
ask dbest k−1
1.11
4.97
23.4
ask+1 dbest k−1
-2.03
-4.24
11.2
ask+2 dbest k−1
-1.60
-1.90
4.44
signk−1
1186
33.3
1
signk−2
753
20.2
1
signk−3
544
14.6
1
signk−4
472
13.4
1
qk−1
4.09
9.29
12.6
qk−2
2.66
6.59
12.6
qk−3
1.85
4.66
12.6
qk−4
1.16
2.98
12.6
bid−2 dbest k−1
Comparable results obtain using as few as two lags of aggressive order sign and signed quantity. Linear forecasts of
k − 4,
yk
do not bene t appreciably from the inclusion of data on aggressive orders before
or on changes in resting depth prior to aggressive order
not normalized by the sign of the
k th
k − 1.
Because the price-change
yk
is
aggressive order, it has an expected value of zero, so I do not
include a constant term in the regression. Including a constant term in the regression has negligible eect on the results. Although the last several aggressive order signs do oer rather remarkable explanatory power, the respective distributions of resting depth changes and executed aggressive order quantities have much heavier tails than the distribution of order sign, so price forecasts are meaningfully improved by the inclusion of these variables. The positive coe cients on the lagged aggressive order variables and on the depth changes at the best bid and best ask are consistent with the general intuition that buy orders portend price increases, and sell orders portend price decreases. The negative coe cients on depth changes at the outside price
58
levels require slightly more explanation. Because the E-mini market operates according to strict price and time priority, a trader who seeks priority execution of his passive order will generally place that order at the best bid (or best ask); however, if the trader believes that an adverse price movement is imminent, he will place his order at the price level that he expects to be the best bid (ask) following the price change. It is relatively uncommon for prices to change immediately after an aggressive order in the E-mini market, but when prices do change, it is extremely rare during regular trading hours for the change to exceed one tick. As a result, the expected best bid (ask) following a price change is typically one tick away from the previous best, so it is not surprising that (e.g.) an increase in resting depth one tick below the best bid tends to precede a downward price change. These features of the E-mini market also shed some light on why changes in depth more than one tick away from the best (i.e., are not signi cant predictors of future price movements.
59
bid−2 dbest k−1
and
ask+2 dbest ) k−1
D Supplemental Tables of Empirical Results
60
61
q¯ explained by regression (18) in excess of that explained by regression (17).
A-HFT vs. All Others denotes the additional explained earnings
45.8
(26.9, 65.1)
3.8
(-7.6, 15.4)
A-HFT vs. Other HFTs
57.7
(39.3, 76.4)
14.5
(3.6, 25.7)
All Others
65.9
(48.0, 85.0)
17.9
(6.9, 29.6)
8.2
(5.5, 10.9)
3.4
q¯ = 5
(1.8, 4.8)
q¯ = 1
A-HFT vs.
A-HFTs
All Others
intervals are reported beneath the point estimates.
(5.3, 42.0)
23.5
(13.9, 50.0)
32.3
(22.1, 57.0)
39.7
(4.1, 10.4)
7.4
q¯ = 10
(13.5, 50.3)
32.3
(25.4, 60.4)
43.2
(35.5, 70.5)
53.3
(7.1, 13.3)
10.1
q¯ = 15
(23.8, 61.8)
42.6
(33.2, 68.6)
51.0
(45.3, 79.9)
62.3
(7.7, 14.7)
11.3
q¯ = 20
the individual A-HFTs. Note that other HFTs includes the other A-HFTs. Units are hundredths of a cent per contract, and
95%
bootstrap con dence
upon the particular A-HFT being excluded, and the reported numbers are averages taken across the slightly dierent groups corresponding to each of
Numbers reported for the A-HFTs are averages over the estimates for individual A-HFTs. The membership of All Others and Other HFTs depends
per contract for a given A-HFT minus those for all other traders, averaged across the A-HFTs; A-HFT vs. Other HFTs denotes an analogous quantity.
greater than
Table 5, below, presents various cross-sectional averages of the estimated additional gross earnings per contract on aggressive orders of submitted size
Table 5: Additional Explained Earnings on Aggressive Orders (Hundredths of a Cent per Contract)
62 45.4
(29.9, 68.4)
48.9
(40.6, 78.1)
59.0
(55.1, 92.0)
73.3
(10.6, 18.4)
14.3
q¯ = 35
(36.2, 77.9)
58.0
(48.3, 87.6)
68.8
(64.6, 103.7)
85.0
(12.0, 20.5)
16.2
q¯ = 40
(35.4, 77.6)
56.0
(49.2, 89.6)
68.8
(65.9, 105.3)
85.0
(11.9, 20.5)
16.2
q¯ = 45
(43.5, 87.1)
64.3
(59.9, 101.9)
80.4
(81.0, 121.9)
100.3
(14.5, 25.3)
19.9
q¯ = 50
(56.2, 99.2)
78.6
(75.5, 117.2)
96.4
(98.5, 138.1)
118.1
(16.2, 27.2)
21.8
q¯ = 60
(53.5, 100.3)
75.7
(68.7, 112.8)
90.2
(86.0, 129.3)
107.3
(11.2, 23.0)
17.1
q¯ = 75
(49.8, 99.1)
74.6
(68.5, 115.1)
92.0
(82.1, 124.2)
104.0
(5.3, 18.7)
12.0
q¯ = 90
traders, or between an A-HFT and all other HFTs.
same (in expectation) for all traders, it has no impact on the
dierence
in additional explained earnings per contract between an A-HFT and all other
The extra explanatory power of (18) re ects the contribution from the private component of information (available to the A-HFT under consideration) A manifested in Ω , as well as the eect of the extra degrees of freedom in (18) relative to (17). Since the eect from the extra degrees of freedom is the
(27.2, 66.0)
36.0
(17.1, 55.4)
A-HFT vs.
Other HFTs
55.1
(37.0, 75.0)
45.0
69.7
(52.0, 89.4)
62.4
(43.7, 80.6)
14.7
(10.6, 18.2)
17.5
(13.5, 21.3)
q¯ = 30
(26.2, 63.9)
All Others
A-HFT vs.
A-HFTs
All Others
q¯ = 25
Table 5Continued: Additional Explained Earnings on Aggressive Orders (Hundredths of a Cent per Contract)
63 4.99 5.28 5.42 5.61 5.87 6.12 6.38 7.65
40
45
50
60
75
90
2000
4.14
20
35
3.85
15
4.79
3.49
10
30
4.23
5
4.41
3.84
1
25
All AOs
q¯
≤ q¯
7.99
7.20
7.20
7.00
6.90
7.04
6.69
7.00
6.79
6.65
4.95
6.39
2.84
4.38
-
AOs
Incremental
A-HFTs
5.67
5.01
5.00
4.98
4.95
4.92
4.91
4.88
4.87
4.81
4.77
4.71
4.66
4.56
4.37
All AOs
≤ q¯
7.63
5.52
5.40
5.30
5.14
5.49
5.14
5.10
5.24
5.14
5.08
4.95
4.85
4.64
-
AOs
Incremental
B-HFTs
3.19
2.03
2.03
1.99
1.96
1.95
1.95
1.91
1.90
1.86
1.83
1.79
1.77
1.72
1.68
All AOs
≤ q¯
4.20
2.16
2.71
2.87
2.00
2.14
2.55
2.40
2.53
2.20
2.06
1.97
1.89
1.75
-
AOs
Incremental
Non-HFTs
Table 6: Average Gross Earnings of Aggressive Orders, by Size and Trader Type (in Dollars per Contract)
64
0.40% 1.44% 3.09% 3.54%
24.31% 43.74% 54.64% 56.75%
1
5
10
15
4.80% 5.37% 6.39% 7.02% 8.51% 9.20% 10.55% 12.97% 16.01% 21.11%
60.82% 62.38% 64.62% 65.82% 68.27% 69.29% 71.07% 73.81% 76.65% 80.68%
20
25
30
35
40
45
50
60
75
90
Volume
% of Aggr.
% of All AOs
q¯
A-HFTs
99.20%
99.02%
98.54%
97.66%
96.32%
96.03%
94.97%
94.14%
92.36%
90.70%
88.01%
84.10%
76.09%
39.48%
% of All AOs
74.86%
73.14%
69.12%
63.29%
55.76%
54.31%
49.56%
46.30%
40.29%
35.82%
30.03%
23.95%
16.13%
4.77%
Volume
% of Aggr.
B-HFTs
97.83%
97.64%
97.36%
97.09%
95.80%
95.66%
95.23%
94.98%
94.41%
93.27%
91.57%
89.87%
83.26%
53.79%
46.37%
44.76%
42.66%
41.08%
34.33%
33.69%
31.94%
31.07%
29.32%
26.43%
23.05%
20.68%
14.88%
5.68%
Volume
% of Aggr.
Non-HFTs
q¯
% of All AOs
Table 7: Fractions of Trader Types' Aggressive Orders Below Size Threshold
